aa r X i v : . [ m a t h . GN ] S e p Asymptotic compactness in topological spaces
Junya Nishiguchi ∗† Abstract
The omega limit sets plays a fundamental role to construct global attractors fortopological semi-dynamical systems with continuous time or discrete time. Therefore,it is important to know when omega limit sets become nonempty compact sets. Thepurpose of this paper is to understand the mechanism under which a given net ofsubsets of topological spaces is compact in the asymptotic sense. For this purpose,we introduce the notion of asymptotic compactness for nets of subsets and study theconnection with the compactness of the limit sets. In this paper, for a given net ofnonempty subsets, we prove that the asymptotic compactness and the property thatthe limit set is a nonempty compact set to which the net converges from above areequivalent in uniformizable spaces. We also study the sequential version of the notionof asymptotic compactness by introducing the notion of sequentiality of directed sets. . Primary 54A20, 54D30, 54E15;Secondary 37B02, 54C60
Keywords . Limit sets; compactness; uniformizable spaces; general topology;set-valued analysis.
Contents ∗ Mathematical Science Group, Advanced Institute for Materials Research (AIMR), Tohoku University,2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan † E-mail: [email protected] Asymptotic compactness and limit set compactness in uniformizablespaces 18
Acknowledgments 28References 28
In the theory of topological semi-dynamical systems with continuous time or discrete time,the omega limit sets is a one of the fundamental objects for the asymptotic behavior ofthe orbits of the semi-dynamical systems (ref. Gottschalk and Hedlund [5, Section 10]).The omega limit sets also plays a fundamental role in the construction of global attractorsin infinite-dimensional semi-dynamical systems (refs. Hale [7], Sell and You [11, Chapter2], and Raugel [10]). Therefore, it is important to understand when omega limit sets arenonempty compact sets when the phase space is neither compact nor locally compact.A related notion for the non-emptiness and the compactness of omega limit sets isknown in the literature. It is the asymptotic compactness introduced in [11] when thephase space of semi-dynamical systems is a metric space. This notion of asymptoticcompactness is expressed by sequences. Therefore, it is not apparent how this notionshould be generalized to the case that the phase space is a general topological spacebecause omega limit points (i.e., points belonging to omega limit sets) are not expectedto be expressed by sequences.In this paper, we will tackle this problem by an approach considering nets ( X s ) s ∈ S of subsets of a topological space X for some directed set S = ( S, ≤ ). Then the limit set L ( X s ) s ∈ S is defined by L ( X s ) s ∈ S = \ s ∈ S cl [ t ∈ S,t ≥ s X t . Here cl denotes the closure operator in the topological space X . For the above mentionedproblem of omega limit sets of topological semi-dynamical systems with continuous timeor discrete time, the directed set S corresponds to the set of nonnegative reals R + or theset of nonnegative integers Z + , respectively. Then the omega limit set ω Φ ( E ) of a subset E ⊂ X for a semiflow Φ : S × X → X is given by the limit set of the net ( Φ ( { s } × E )) s ∈ S of subsets of X . Here a semiflow Φ is a map with the properties that (i) Φ (0 , x ) = x forall x ∈ X and (ii) Φ ( t + s, x ) = Φ ( t, Φ ( s, x )) for all t, s ∈ S and all x ∈ X .This approach is general from the perspective on considering omega limit sets oftopological semi-dynamical systems with continuous time or discrete time. At the same2ime, this approach should be natural because the nets ( X s ) s ∈ S contains nets of pointsin X , which are necessary to consider the convergence concept in general topologicalspaces. Furthermore, this approach permits us to study omega limit sets of topologicalsemi-dynamical systems with a preordered abelian phase group T = ( T, + , ≤ ) (i.e., anabelian phase group with translation-invariant preorder) because the positive cone givenby T + := { t ∈ T : t ≥ } becomes a directed set. For example, see also [6] for a study ofthis direction.For the net ( X s ) s ∈ S of subsets of X , we will obtain characterizations of points belongingto the limit set L ( X s ) s ∈ S . Then we will define the notions of asymptotic compactness and weak asymptotic compactness of the net ( X s ) s ∈ S based on these characterizations.See Definitions 3.1 and 3.3 for further details. These are the key notions of this paper,which are expected to be related with the compactness of the limit set L ( X s ) s ∈ S . Wewill also introduce the convergence concepts for the net ( X s ) s ∈ S called the convergencefrom above and the convergence from below based on the upper semicontinuity and thelower semicontinuity of set-valued maps because ( X s ) s ∈ S is a set-valued map from S to X . See Definitions 2.25 and 2.34 for their definitions. To make clear the above mentionedconnection, we will introduce the terminology of the limit set compactness of ( X s ) s ∈ S ,which means that L ( X s ) s ∈ S is a nonempty compact set to which ( X s ) s ∈ S converges fromabove.Unfortunately, it seems to be optimistic to expect that the asymptotic compactnessand the limit set compactness are equivalent in any general topological space. One of thereason is that even if we choose a net ( z α ) α ∈ A for some directed set A in the limit set L ( X s ) s ∈ S , we cannot associate this net with a net in general which is related to the net( X s ) s ∈ S . Then we cannot extract the full power of the asymptotic compactness.To overcome this difficulty, we rely on the uniformizability of the topological space X .Under the assumption of the uniformizabiity, for a given net ( z α ) α ∈ A in L ( X s ) s ∈ S , we canchoose a net ( y β ) β ∈ B for some directed set B which is related to the net ( X s ) s ∈ S so that z α and y β are in some uniform nearness. Here the uniformizability is essentially used. Themathematically precise statement is given in the proof of Theorem 4.6.The final task of this paper is to clarify the relation between the asymptotic compact-ness of the net ( X s ) s ∈ S and the asymptotic compactness for semiflows with continuoustime in metric spaces introduced in [11]. For this purpose, we will introduce the sequen-tial versions of the asymptotic compactness and the weak asymptotic compactness (seeDefinitions 5.7 and 5.10). To introduce the sequential versions, we need to restrict a classof directed sets. This is a class of sequential directed sets , in which a directed set has asequence ( s n ) ∞ n =1 which becomes larger and larger as n → ∞ . See Definition 5.1 for theprecise definition. Then we will obtain the equivalence between the asymptotic compact-ness and the asymptotic sequential compactness for the net ( X s ) s ∈ S when S is sequentialand the topological space X is pseudo-metrizable. This shows the above mentioned equiv-alence.This paper is organized as follows. In Section 2, we study the limit set and theconvergence property for a given net ( X s ) s ∈ S of subsets of X for some directed set S . Thenwe obtain characterizations of points belonging to the limit set L ( X s ) s ∈ S in Theorem 2.15.We also study various connections between the convergence of the net ( X s ) s ∈ S and thelimit set L ( X s ) s ∈ S . In Section 3, we introduce the notions of asymptotic compactnessand weak asymptotic compactness of the net ( X s ) s ∈ S of subsets in Definitions 3.1 and3.3. In Lemma 3.15, we reveal the connection between the asymptotic compactness and3he limit set compactness of ( X s ) s ∈ S when X is a locally compact regular space by usingthe notion of eventual Lagrange stability. In Section 4, we investigate the asymptoticcompactness of nets of subsets in uniformizable spaces. One of the main result of thispaper is Theorem 4.7, which shows that the following properties are equivalent when X is uniformizable: (i) ( X s ) s ∈ S converges from above to some nonempty compact set, (ii)( X s ) s ∈ S is asymptotically compact, and (iii) ( X s ) s ∈ S is limit set compact. In Section 5,we study the sequential versions of the asymptotic compactness and the weak asymptoticcompactness. Under the assumption that S is sequential and X is pseudo-metrizable,we obtain the sequential version of Theorem 4.7 in Theorem 5.19. By combining thesetheorems, we finally obtain the equivalence between the asymptotic compactness and theasymptotic sequential compactness in Corollary 5.20. Throughout this section, let X be a topological space. In this subsection, we will investigate characterizations of points belonging to the limitset of a net of subsets of X . For this purpose, we first recall the convergence concepts intopological spaces via nets. We refer the reader to [8] as a general reference of generaltopology. (ref. [8]) . A nonempty set A together with a preorder ≤ on A (i.e., abinary relation on A with the reflexivity and the transitivity) is called a directed set ifevery pair of a, b ∈ A has an upper bound c ∈ A , i.e., an element c ∈ A satisfying a ≤ c and b ≤ c . A directed set ( A, ≤ ) is called a directed poset if ≤ is a partial order, i.e., ≤ satisfies the antisymmetry. Remark . For each directed sets A = ( A, ≤ ) and B = ( B, ≤ ), the Cartesian product A × B is considered as a directed set with the preorder ≤ defined as follows: ( α , β ) ≤ ( α , β ) if α ≤ α and β ≤ β . The directed set ( A × B, ≤ ) is called the product directedset . Definition 2.3 (ref. [8]) . A family ( x α ) α ∈ A in some set for some directed set A is calleda net . Definition 2.4 (ref. [8]) . Let A = ( A, ≤ ) be a directed set and U ⊂ X be a subset. • A net ( x α ) α ∈ A in X is said to be eventually in U if there exists α ∈ A such that forall α ∈ A , α ≥ α implies x α ∈ U . • A net ( x α ) α ∈ A in X is said to be frequently in U if for every α ∈ A , there exists β ∈ A such that β ≥ α and x β ∈ U . Definition 2.5 (ref. [8]) . Let A = ( A, ≤ ) be a directed set and x ∈ X be given. • A net ( x α ) α ∈ A in X is said to converge to x if ( x α ) α ∈ A eventually in every neigh-borhood of x . 4 A net ( x α ) α ∈ A in X is said to have a cluster point x if ( x α ) α ∈ A frequently in everyneighborhood of x . Definition 2.6 (ref. [8]) . Let A = ( A, ≤ ) be a directed set. A subset S ⊂ A is said to be cofinal if for every α ∈ A , there exists α ′ ∈ S such that α ′ ≥ α . Remark . A subset S of the directed set A is not necessarily directed. We note thatany cofinal subset of A is directed. Definition 2.8.
Let A = ( A, ≤ ) be a directed set and B be a set. A map h : B → A issaid to be final if the image h ( B ) := { h ( β ) : β ∈ B } is a cofinal subset of A .In this paper, we adopt the following notation. Notation . Let A = ( A, ≤ ) , B = ( B, ≤ ) be directed sets and h : B → A be a map. By h ( β ) → bd( A ) as β → bd( B ), we mean that for every α ∈ A , there exists β ∈ B suchthat for all β ∈ B , β ≥ β implies h ( β ) ≥ α .We note that a map h : A → A satisfying h ( α ) ≥ α for all α ∈ A has the property h ( α ) → bd( A ) as α → bd( A ). Definition 2.9 (ref. [8]) . Let ( x α ) α ∈ A be a net in X for some directed set A . For everydirected set B , a net ( y β ) β ∈ B is called a subnet of ( x α ) α ∈ A if there exists a map h : B → A such that (i) h ( β ) → bd( A ) as β → bd( B ) and (ii) y β = x h ( β ) holds for all β ∈ B .In this paper, we do not adopt the convention that the map h : B → A is monotoneand final for a subnet ( x h ( β ) ) β ∈ B . The following theorem gives characterizations of clusterpoints of nets. See [8] for the proof. See also Theorem 2.15 for an extension of thischaracterizations to nets of subsets. Theorem 2.10 (ref. [8]) . Let A be a directed set, ( x α ) α ∈ A be a net in X , and x ∈ X begiven. Then the following properties are equivalent: (a) x is a cluster point of ( x α ) α ∈ A . (b) There exist a directed set B and a monotone final map h : B → A such that ( x h ( β ) ) β ∈ B converges to x . (c) There exist a directed set B and a map h : B → A such that h ( β ) → bd( A ) as β → bd( B ) and ( x h ( β ) ) β ∈ B converges to x .Notation . For each x ∈ X , let N x denote the set of all neighborhoods of x . It isconsidered to be a directed poset with the partial order ≤ defined as follows: For all U , U ∈ N x , U ≤ U if U ⊃ U . Notation . For each subset E ⊂ X , let cl( E ) denote the closure of E (i.e., the smallestclosed set of X containing E ). Then x ∈ cl( E ) if and only of U ∩ E = ∅ holds for every U ∈ N x . Remark . Let A = ( A, ≤ ) be a directed set and ( x α ) α ∈ A be a net in X . Then x ∈ T α ∈ A cl { x β : β ∈ A, β ≥ α } if and only if for every α ∈ A and every U ∈ N x , U ∩ { x β : β ∈ A, β ≥ α } 6 = ∅ . Therefore, \ α ∈ A cl { x β : β ∈ A, β ≥ α } is equal to the set of cluster points of ( x α ) α ∈ A .5he following is another characterization of the points belonging to the closure of anonempty subset. Lemma 2.12.
Let E ⊂ X be a nonempty subset and x ∈ X be given. Then x ∈ cl( E ) ifand only if there exist a directed set A and a net ( x α ) α ∈ A in E such that ( x α ) α ∈ A convergesto x . The proof is standard, and therefore, it can be omitted.
Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of subsets of X .The subset L ( X s ) s ∈ S ⊂ X defined by L ( X s ) s ∈ S = \ s ∈ S cl [ t ∈ S,t ≥ s X t is called the limit set of ( X s ) s ∈ S . Remark . When S = R + or Z + (the set of nonnegative real numbers or the set ofnonnegative integers) and X s = Φ ( { s } × E ) for some semiflow Φ : S × X → X and forsome subset E ⊂ X , the limit set L ( X s ) s ∈ S is called the omega limit set of E for Φ . Itwill be denoted by ω Φ ( E ).The above limit set should be distinguished from the set-theoretic limit set. We notethat from Remark 2.11, the limit set is a generalization of the set of cluster points for netsof points. The following theorem gives characterizations of points belonging to a limit set.It is considered to be a generalization of Theorem 2.10. Theorem 2.15.
Let S = ( S, ≤ ) be a directed set, ( X s ) s ∈ S be a net of subsets of X , and y ∈ X be given. Then the following properties are equivalent: (a) y ∈ L ( X s ) s ∈ S . (b) For every directed set I and every subnet ( X s i ) i ∈ I of ( X s ) s ∈ S , there exist a directedset J , a monotone final map h : J → I , and ( y j ) j ∈ J ∈ Q j ∈ J S t ≥ s h ( j ) X t such thatthe net ( y j ) j ∈ J in X converges to y . (c) There exist a directed set I , a monotone final map h : I → S , and ( y i ) i ∈ I ∈ Q i ∈ I S t ∈ S,t ≥ h ( i ) X t such that the net ( y i ) i ∈ I in X converges to y . (d) There exist a directed set I , a subnet ( X s i ) i ∈ I of ( X s ) s ∈ S , and ( y i ) i ∈ I ∈ Q i ∈ I X s i such that the net ( y i ) i ∈ I in X converges to y .Proof. (a) ⇒ (b): Let I = ( I, ≤ ) be a directed set and ( X s i ) i ∈ I be a subnet of ( X s ) s ∈ S .We consider the subset J of the product directed set I × N y given by J := n ( i, U ) ∈ I × N y : S t ≥ s i X t ∩ U = ∅ o .
6y the definition of L ( X s ) s ∈ S , J is directed. We can choose a net ( y i,U ) ( i,U ) ∈ J so that y i,U ∈ [ t ≥ s i X t ∩ U for all ( i, U ) ∈ J . By defining a monotone final map h : J → I by h ( i, U ) = i (( i, U ) ∈ J ) , it holds that y i,U ∈ S t ≥ s h ( i,U ) X t for all ( i, U ) ∈ J and ( y i,U ) ( i,U ) ∈ J converges to y .(b) ⇒ (c): This is obvious.(c) ⇒ (d): For each i ∈ I , there exists s i ∈ S such that s i ≥ h ( i ) and y i ∈ X s i . Since s i → bd( S ) as i → bd( I ), (d) holds.(d) ⇒ (a): Let s ∈ S be fixed. Then there is i ∈ I such that for all i ∈ I , i ≥ i implies s i ≥ s . Let I := { i ∈ I : i ≥ i } be a directed subset of I . Since ( y i ) i ∈ I is a netin S t ∈ S,t ≥ s X t converging to y , we have y ∈ cl [ t ∈ S,t ≥ s X t from Lemma 2.12. This shows y ∈ L ( X s ) s ∈ S .We now compare the limit set introduced in Definition 2.13 with the upper and lowerlimit of sequences of subsets of (pseudo-) metric spaces. Notation . Let d be a pseudo-metric on X . For each x ∈ X and each subset A ⊂ X , let d ( x, A ) := inf y ∈ A d ( x, y ) . We interpret that d ( x, ∅ ) is equal to ∞ . Then d ( x, A ) < ∞ if and only if A is nonempty.For each nonempty subsets A, B ⊂ X , let d ( A ; B ) := sup x ∈ A inf y ∈ B d ( x, y ) = sup x ∈ A d ( x, B )We promise that d ( A ; ∅ ) = ∞ for any nonempty subset A and d ( ∅ ; B ) = 0 for anynonempty subset B . We note that d ( ∅ ; ∅ ) is not defined. Remark . Let X = ( X, d ) be a pseudo-metric space. For each subset A ⊂ X , thefunction d ( · , A ) : X → R satisfies the following properties: • d ( x, A ) = 0 if and only if x ∈ cl( A ). • For all x, y ∈ X , | d ( x, A ) − d ( y, A ) | ≤ d ( x, y ).In particular, the function d ( · , A ) : X → R is continuous. Definition 2.17 (ref. [2]) . Let X = ( X, d ) be a pseudo-metric space and ( X n ) ∞ n =1 be asequence of subsets of X . The upper limit and the lower limit are defined byLim sup n →∞ X n = n y ∈ X : lim inf n →∞ d ( y, X n ) = 0 o , Lim inf n →∞ X n = n y ∈ X : lim n →∞ d ( y, X n ) = 0 o , respectively. Here lim inf n →∞ d ( y, X n ) := sup m ≥ inf n ≥ m d ( y, X n ).7n this paper, a subsequence of some sequence ( x n ) ∞ n =1 means a subnet ( x n k ) ∞ k =1 (see[8]). The following are characterizations of the upper limit and the lower limit of a givensequence of subsets of pseudo-metric spaces. Lemma 2.18 (cf. [2]) . Let X = ( X, d ) be a pseudo-metric space, ( X n ) ∞ n =1 be a sequenceof subsets of X , and y ∈ X be given. Then the following properties are euivalent: (a) y ∈ Lim sup n →∞ X n . (b) There exist a subsequence ( X n k ) ∞ k =1 of ( X n ) ∞ n =1 and ( y k ) ∞ k =1 ∈ Q ∞ k =1 X n k such thatthe sequence ( y k ) ∞ k =1 in X converges to y .Proof. (a) ⇒ (b): By definition, inf n ≥ k d ( y, X n ) = 0holds for all k ≥
1. Therefore, for each given k ≥
1, there are an integer n k ≥ k and y k ∈ X n k such that d ( y, y k ) < k . This shows that (b) holds.(b) ⇒ (a): Let ε > k ≥ k ≥ k , d ( y, y k ) < ε holds. Let m ≥ k ≥ k so that n k ≥ m . Since inf n ≥ m d ( y, X n ) ≤ d ( y, X n k ) ≤ d ( y, y k ) < ε, we have lim inf n →∞ d ( y, X n ) = sup m ≥ inf n ≥ m d ( y, X n ) ≤ ε. This holds for arbitrary ε >
0, and therefore, we have lim inf n →∞ d ( y, X n ) = 0. Lemma 2.19 (cf. [2]) . Let X = ( X, d ) be a pseudo-metric space, ( X n ) ∞ n =1 be a sequenceof subsets of X , and y ∈ X be given. Then the following properties are equivalent: (a) y ∈ Lim inf n →∞ X n . (b) There exist an integer n ≥ and ( x n ) ∞ n = n ∈ Q ∞ n = n X n such that the sequence ( x n ) ∞ n = n in X converges to y .Proof. (a) ⇒ (b): Since d ( y, X n ) → n → ∞ , there is n ≥ d ( y, X n ) < ∞ for all n ≥ n . We can choose a sequence ( x n ) ∞ n = n in X so that x n ∈ X n and d ( y, x n ) < d ( y, X n ) + 1 n hold for all n ≥ n . Then the sequence ( x n ) ∞ n = n converges to y because d ( y, X n ) → n → ∞ .(b) ⇒ (a): This is obvious because d ( y, X n ) ≤ d ( y, x n ) holds for all n ≥ n . Remark . In Lemma 2.19, we can choose n = 1 when each X n is nonempty. In [2],this non-emptiness is implicitly assumed, where it is stated that y ∈ Lim sup n →∞ X n ifand only if y is a cluster point of some sequence ( x n ) ∞ n =1 belonging to Q ∞ n =1 X n .8y combining Theorem 2.15 and Lemma 2.18, the following statement is obtained asa corollary. The proof can be omitted. We note that it is also mentioned in [2]. Corollary 2.21.
Let X = ( X, d ) be a pseudo-metric space and ( X n ) ∞ n =1 be a sequence ofsubsets of X . Then L ( X n ) ∞ n =1 = Lim sup n →∞ X n holds. To introduce convergence concepts for nets of subsets of X , we recall the definitions ofupper and lower semicontinuity of set-valued maps. Definition 2.22 (ref. [1]) . Let A and B be sets. A map F : A → B is called a set-valuedmap from A to B . Here 2 B denotes the set of all subsets of B . The set-valued map F isalso denoted by F : A ⇒ B . For each subset A ⊂ A , let F ( A ) := S a ∈ A F ( a ). (ref. [1]) . Let Y be a topological space, F : X ⇒ Y be a set-valued map,and x ∈ X be given. F is said to be upper semicontinuous at x if for every neighborhood U of F ( x ), there exists V ∈ N x such that F ( V ) ⊂ U holds. Remark . When x is an isolated point of X (i.e., { x } is an open set of X ), the set-valued map F is always upper semicontinuous at x .Based on this definition, we introduce the following. Definition 2.25.
Let S = ( S, ≤ ) be a directed set, ( X s ) s ∈ S be a net of subsets of X ,and A ⊂ X be a subset. We say that ( X s ) s ∈ S converges from above to A if for everyneighborhood U of A , there exists s ∈ S such that S t ∈ S,t ≥ s X t ⊂ U holds. Remark . Let Y be a topological space, F : X ⇒ Y be a set-valued map, and x ∈ X be given. For every V , V ∈ N x , V ≤ V implies F ( V ) ⊃ F ( V ). Therefore, F is uppersemicontinuous at x if and only if the net ( F ( V )) V ∈N x of subsets converges from above to F ( x ).In the rest of this subsubsection, we study the property of the convergence from abovein pseudo-metric spaces. Notation . Let d be a pseudo-metric on X . For each x ∈ X , each subset A ⊂ X , andeach r >
0, let B d ( x ; r ) := { y ∈ X : d ( x, y ) < r } , B d ( A ; r ) := [ x ∈ A B d ( x ; r ) . Remark . Let X = ( X, d ) be a pseudo-metric space, S = ( S, ≤ ) be a directed set, and( X s ) s ∈ S be a net of subsets of X . Then the following properties are equivalent: • For every ε >
0, there exists s ∈ S such that for all s ≥ s , X s ⊂ B d ( A ; ε ) holds. • d ( X s ; A ) → s → bd( S ).The following is a key lemma to study the property of the convergence from above inpseudo-metric spaces. We give a proof for the sake of completeness although its statementis mentioned in [1, page 45 and page 66]. 9 emma 2.28. Suppose that X = ( X, d ) is a pseudo-metric space. Let K ⊂ X be anonempty compact set. Then for every neighborhood U of K , there exists δ > such that B d ( K ; δ ) ⊂ U holds.Proof. We only have to consider the case that U is open. Let r : X → R be the continuousfunction defined by r ( x ) = d ( x, X \ U ) ( x ∈ X ) . Then for every x ∈ X , d ( x, y ) < r ( x ) implies y ∈ U . Furthermore, r ( x ) > x ∈ K since X \ U is closed. The extreme value theorem ensures the existence of x ∈ K satisfying r := r ( x ) = inf x ∈ K r ( x ) . This implies r >
0, and we have B d ( K ; r ) ⊂ [ x ∈ K B d ( x ; r ( x )) ⊂ U. This shows the conclusion.We can obtain the following corollary by using Lemma 2.28.
Corollary 2.29.
Suppose that X = ( X, d ) is a pseudo-metric space. Let S = ( S, ≤ ) be adirected set, ( X s ) s ∈ S be a net of subsets of X , and K ⊂ X be a nonempty compact set.Then the following properties are equivalent: (a) ( X s ) s ∈ S converges from above to K . (b) lim s → bd( S ) d ( X s ; K ) = 0 , i.e., the net ( d ( X s ; K )) s ∈ S of nonnegative real numbersconverges to . When the above property (b) holds, we will say that ( X s ) s ∈ S converges from above to A with respect to d . Remark . Let X = ( X, d ) be a metric space, S = R + or Z + , and Φ : S × X → X be a semiflow. A nonempty set A ⊂ X is said to attract a subset E ⊂ X under Φ if thenet ( Φ ( { s } × E )) s ∈ S converges from above to A with respect to d . (cf. [1, 2]) . Let Y be a topological space, F : X ⇒ Y be a set-valuedmap, and x ∈ X be given. F is said to be lower semicontinuous at x if for every y ∈ F ( x )and every U ∈ N y , there exists V ∈ N x such that for all x ′ ∈ V , F ( x ′ ) ∩ U = ∅ holds. Weinterpret that F is lower semicontinuous at x when F ( x ) is empty. Remark . In [1, Definition 2 in Chapter 1] or [2, Definition 1.4.2], it is assumed that F ( x ) is always nonempty for every x ∈ X , or F is restricted to the domain of definition defined by { x ∈ X : F ( x ) = ∅} . Remark . In the same way as the upper semicontinuity, the set-valued map F is alwayslower semicontinuous at x when x is an isolated point of X .Based on this definition, we introduce the following convergence concept.10 efinition 2.34. Let S = ( S, ≤ ) be a directed set, ( X s ) s ∈ S be a net of subsets of X , and A ⊂ X be a given subset. We say that ( X s ) s ∈ S converges from below to A if for every y ∈ A and every U ∈ N y , there exists s ∈ S such that for all s ∈ S , s ≥ s implies X s ∩ U = ∅ . We also say that ( X s ) s ∈ S converges to A if ( X s ) s ∈ S converges from aboveand from below to A .By definition, when ( X s ) s ∈ S is single-valued, i.e., X s = { x s } holds for all s ∈ S forsome net ( x s ) s ∈ S in X , the convergence from below of ( X s ) s ∈ S to some subset A impliesthe convergence from above of ( X s ) s ∈ S to A . Notation . Suppose that X = ( X, d ) is a pseudo-metric space. Let x ∈ X be given.We consider the binary relation ≤ on X \ { x } defined as follows: x ≤ x if d ( x, x ) ≥ d ( x, x ). Then ≤ becomes a preorder. Furthermore, if x is not an isolated point of X ,then X \ { x } = ( X \ { x } , ≤ ) becomes a directed set. Remark . Suppose that X = ( X, d ) is a pseudo-metric space. Let Y be a topologicalspace, F : X ⇒ Y be a set-valued map, and x ∈ X be not an isolated point. For any x ′ ∈ X \ { x } , x ′ ≥ x for some x ∈ X \ { x } is equivalent to the property that x ′ belongsto the closed ball ¯ B d ( x ; d ( x ; x )) := { y ∈ X : d ( x, y ) ≤ d ( x, x ) } . Therefore, F is lowersemicontinuous at x if and only if the net ( F ( x ′ )) x ′ ∈ X \{ x } of subsets converges from belowto F ( x ).The following is a generalization of the characterization of the continuity of mapsbetween topological spaces in terms of nets of points. Theorem 2.36 (cf. [1]) . Let Y be a topological space, F : X ⇒ Y be a set-valued map,and x ∈ X . Then the following properties are equivalent: (a) F is lower semicontinuous at x . (b) For every directed set A = ( A, ≤ ) and every net ( x α ) α ∈ A in X converging to x , thenet ( F ( x α )) α ∈ A of subsets converges from below to F ( x ) . (c) For every y ∈ F ( x ) , every directed set A = ( A, ≤ ) , and every net ( x α ) α ∈ A in X converging to x , there exists a subnet ( x α β ) β ∈ B for some directed set B = ( B, ≤ ) such that ( y β ) β ∈ B ∈ Q β ∈ B F ( x α β ) and the net ( y β ) β ∈ B in X converges to y .Proof. We only have to consider the case that F ( x ) is nonempty.(a) ⇒ (b): Let A = ( A, ≤ ) be a directed set and ( x α ) α ∈ A be a net in X convergingto x . Let y ∈ F ( x ) and U ∈ N y be given. Since F is lower semicontinuous at x , there is V ∈ N x such that for all x ′ ∈ V , F ( x ′ ) ∩ U = ∅ holds. Then there is α ∈ A such that forall α ∈ A , α ≥ α implies x α ∈ V , which implies F ( x α ) ∩ U = ∅ for such α . Therefore, (b)holds.(b) ⇒ (c): Let y ∈ F ( x ), A = ( A, ≤ ) be a directed set, and ( x α ) α ∈ A be a net in X converging to x . Let B be the subset of the product directed set A × N y given by B := { ( α, U ) ∈ A × N y : F ( x α ) ∩ U = ∅} . By the assumption, B is nonempty and directed. Then we can choose a net ( y α,U ) ( α,U ) ∈ B in X so that y α,U ∈ F ( x α ) ∩ U α, U ) ∈ B . By this choice, the net ( y α,U ) ( α,U ) ∈ B converges to y . We consider themonotone final map h : B → A defined by h ( α, U ) = α (( α, U ) ∈ B ) . Then we have y α,U ∈ F ( x h ( α,U ) ) for every ( α, U ) ∈ B . Therefore, (c) holds.(c) ⇒ (a): We suppose the contrary and derive a contradiction. Then there are y ∈ F ( x ) and an open neighborhood U of x with the following property: For every V ∈ N x , there is x V ∈ V such that F ( x V ) ∩ U = ∅ . Since the net ( x V ) V ∈N x converges to x , there is a subnet ( x V β ) β ∈ B for some directed set B such that ( y β ) β ∈ B ∈ Q β ∈ B F ( x V β )and the net ( y β ) β ∈ B converges to y . This ( y β ) β ∈ B is a net in X \ U because F ( x V ) ∩ U = ∅ for all V ∈ N x . Therefore, we have y ∈ X \ U from Lemma 2.12, which is a contradiction.Thus, (a) holds. Remark . In [1], it is stated that F is lower semicontinuous at x if and only if forany y ∈ F ( x ), any directed set A , and any net ( x α ) α ∈ A in X converging to x , there exists( y α ) α ∈ A ∈ Q α ∈ A F ( x α ) such that the net ( y α ) α ∈ A converges to y .The following theorem gives a characterization of the property of the convergence frombelow. The proof is similar to that of Theorem 2.36, but there is a slight difference. Wegive a proof for the sake of completeness. Theorem 2.38.
Let S = ( S, ≤ ) be a directed set, ( X s ) s ∈ S be a net of subsets of X , and A ⊂ X be a subset. Then the following properties are equivalent: (a) ( X s ) s ∈ S converges from below to A . (b) The following statement holds for each y ∈ A : For every directed set I and everysubnet ( X s i ) i ∈ I of ( X s ) s ∈ S , there exist a directed set J , monotone final map h : J → I , and ( y j ) j ∈ J ∈ Q j ∈ J X s h ( j ) such that the net ( y j ) j ∈ J in X converges to y . (c) The following statement holds for each y ∈ A : For every directed set I and everysubnet ( X s i ) i ∈ I of ( X s ) s ∈ S , there exist a directed set J , a subnet ( s i j ) j ∈ J of ( s i ) i ∈ I ,and ( y j ) j ∈ J ∈ Q j ∈ J X s ij such that the net ( y j ) j ∈ J in X converges to y .Proof. We only have to consider the case that A is nonempty.(a) ⇒ (b): Let y ∈ A , I = ( I, ≤ ) be a directed set, and ( X s i ) i ∈ I be a subnet of ( X s ) s ∈ S .We consider the subset J of the product directed set I × N y given by J := { ( i, U ) ∈ I × N y : X s i ∩ U = ∅} . Step 1.
We claim that J is a directed set. We first show that J is nonempty. Wechoose some U ∈ N y . By the assumption, there is t U ∈ S such that for all s ∈ S , s ≥ t U implies X s ∩ U = ∅ . Since s i → bd( S ) as i → bd( I ), there is i U ∈ I such that for all i ∈ I , i ≥ i U implies s i ≥ t U . This means ( i, U ) ∈ J for all i ≥ i U , and therefore, J is nonempty. We next show that J = ( J, ≤ ) is directed.Let ( i , U ) , ( i , U ) ∈ J be given. Let U := U ∩ U ∈ N y . By choosing an upper bound12 ∈ I of i , i , i U , we obtain an upper bound ( i, U ) ∈ J of ( i , U ) , ( i , U ). Therefore, J isdirected. Step 2.
We can choose a net ( y i,U ) ( i,U ) ∈ J so that y i,U ∈ X s i ∩ U for all ( i, U ) ∈ J . By defining a monotone final map h : J → I by h ( i, U ) = i (( i, U ) ∈ J ) , it holds that y i,U ∈ X s h ( i,U ) for all ( i, U ) ∈ J and ( y i,U ) ( i,U ) ∈ J converges to y . Therefore,(b) holds.(b) ⇒ (c): This is obvious.(c) ⇒ (a): We suppose the contrary and derive a contradiction. Then there are y ∈ A and an open neighborhood U of y with the following property: For every i ∈ S , there is s i ∈ S such that s i ≥ i and X s i ∩ U = ∅ . Since s i → bd( S ) as i → bd( S ), ( X s i ) i ∈ S is a subnet of ( X s ) s ∈ S . Then we can choose adirected set J , a subnet ( s i j ) j ∈ J of ( s i ), and a net ( y j ) j ∈ J so that y j ∈ X s ij for all j ∈ J and ( y j ) j ∈ J converges to y . This ( y j ) j ∈ J is a net in X \ U because X s i ⊂ X \ U holds for all i ∈ S . Therefore, we have y ∈ X \ U from Lemma 2.12, which is a contradic-tion. Thus, (a) holds.By combining Theorems 2.15 and 2.38, we obtain the following corollary. The proofcan be omitted. Corollary 2.39.
Let S be a directed set, ( X s ) s ∈ S be a net of subsets of X , and A ⊂ X bea given subset. If ( X s ) s ∈ S converges from below to A , then A ⊂ L ( X s ) s ∈ S holds. In the rest of this subsubsection, we study the property of the convergence from belowin pseudo-metric spaces.
Remark . Let X = ( X, d ) be a pseudo-metric space, S = ( S, ≤ ) be a directed set,( X s ) s ∈ S be a net of subsets of X , and y ∈ X be given. Then the following properties areequivalent: • For every ε >
0, there exists s ∈ S such that for all s ≥ s , X s ∩ B d ( y ; ε ) holds. • lim s → bd( S ) d ( y, X s ) = 0.In view of the above remark, d ( A ; X s ) → s → bd( S ) is sufficient for the conver-gence from below of ( X s ) s ∈ S to A . As the following theorem shows, the condition is alsonecessary when A is compact. Theorem 2.41.
Suppose that X = ( X, d ) is a pseudo-metric space. Let S = ( S, ≤ ) be adirected set, ( X s ) s ∈ S be a net of subsets of X , and K ⊂ X be a nonempty compact set.Then ( X s ) s ∈ S converges from below to K if and only if lim s → bd( S ) d ( K ; X s ) = 0 . The proof is similar to that of the following corresponding theorem about the lowersemicontinuity of set-valued maps. Therefore, it can be omitted.13 heorem 2.42 (ref. [1]) . Let Y = ( Y, ρ ) be a pseudo-metric space, F : X ⇒ Y be aset-valued map, and x ∈ X be given. Suppose that F ( x ) is compact. Then F is lowersemicontinuous at x if and only if lim x ′ → x ρ ( F ( x ); F ( x ′ )) = 0 holds. By lim x ′ → x ρ ( F ( x ); F ( x ′ )) = 0, we mean that for every ε >
0, there exists V ∈ N x suchthat for all x ′ ∈ V , ρ ( F ( x ); F ( x ′ )) < ε holds. See [1, Proof of Proposition 3 in Chapter 1]for the proof. In this subsection, we will investigate the connection between the convergence propertyand limit sets under the separation axioms.
For the purpose stated above, we recall the separation axioms for topological spaces.
Definition 2.43 (ref. [8]) . X is said to be Hausdorff if for every x, y ∈ X satisfying x = y , there exist U ∈ N x and V ∈ N y such that U ∩ V = ∅ . Theorem 2.44 (ref. [8]) . Suppose that X is Hausdorff and let K ⊂ X be a compact set.Then for every y ∈ X \ K , there are neighborhood U of K and a neighborhood V of y suchthat U ∩ V = ∅ . Here a topological space is said to be compact if every open cover has a finite subcover(ref. [8]). The compactness can be paraphrased by using the finite intersection property.
Definition 2.45 (ref. [8]) . Let C be a collection of subsets of X . C is said to have the finite intersection property if T C := T C ∈C C is nonempty for every finite sub-collection C ⊂ C . Lemma 2.46 (ref. [8]) . X is compact if and only if for every collection of closed sets of X with the finite intersection property, the intersection is nonempty. Then the compactness can be characterized by the existence of a convergent subnet.
Theorem 2.47 (ref. [8]) . X is compact if and only if every net in X has a cluster point.Consequently, X is compact if and only if every net in X has a convergent subnet. Definition 2.48 (ref. [8]) . X is said to be regular if for every x ∈ X and every U ∈ N x ,there exists a closed neighborhood V of x contained in U . Remark . X is regular if and only if for every closed set F of X and every x F ,there exist open sets U and V of X such that x ∈ U , F ⊂ V , and U ∩ V = ∅ . (cf. [3]) . Suppose that X is Hausdorff. Let S = ( S, ≤ ) be a directed setand ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S converges from above to some compactset K ⊂ X , then L ( X s ) s ∈ S ⊂ K holds. Consequently, if ( X s ) s ∈ S converges to K , then L ( X s ) s ∈ S = K holds. roof. We suppose L ( X s ) s ∈ S K and derive a contradiction. Then we can choose y ∈ X so that y ∈ L ( X s ) s ∈ S ∩ ( X \ K ) . From Theorem 2.44, there are a neighborhood U of K and a neighborhood V of y suchthat U ∩ V = ∅ . For this U , there is s ∈ S such that S t ∈ S,t ≥ s X t ⊂ U . Therefore, we have [ t ∈ S,t ≥ s X t ∩ V ⊂ U ∩ V = ∅ , which implies y cl S t ∈ S,t ≥ s X t . This contradicts y ∈ L ( X s ) s ∈ S . L ( X s ) s ∈ S = K underthe convergence of ( X s ) s ∈ S to K is a consequence of Corollary 2.39. Remark . Let Φ : R + × X → X be a semiflow and E, A ⊂ X be subsets. A is said to attract E under Φ if the net ( Φ ( { s } × E )) s ∈ R + of subsets of X converges from above to A .We note that this is not equivalent to the attraction in metric spaces. In [3, Proposition 2.3in Chapter XI], it is proved that if A is compact and attracts E under Φ , then ω Φ ( E ) ⊂ A holds. Lemma 2.52 (cf. [9]) . Suppose that X is regular. Let S = ( S, ≤ ) be a directed setand ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S converges from above to some subset A ⊂ X , then L ( X s ) s ∈ S ⊂ cl( A ) holds. Consequently, if ( X s ) s ∈ S converges to A , then L ( X s ) s ∈ S = cl( A ) holds. We omit the proof because the argument is similar to the proof of Lemma 2.50. Seealso [9, Proof of Theorem 2.10].
Remark . Suppose that X is a regular Hausdorff space. Let Φ : R + × X → X be asemiflow and E, A ⊂ X be subsets. In [9, Theorem 2.10], it is proved that if A attracts E under Φ , then ω Φ ( E ) ⊂ cl( A ) holds. Throughout this section, let X be a topological space. The purpose of this section is tointroduce the notions of asymptotic compactness and weak asymptotic compactness andinvestigate their fundamental properties. We also study their connection with the limitset compactness and the eventual Lagrange stability introduced below. In view of Theorem 2.15, we introduce the following notions of asymptotic compactnessfor nets of nonempty subsets. They are considered to be generalizations of the asymptoticcompactness of subsets under semiflows with continuous time in metric spaces (see [11]).See also Definition 5.7.
Definition 3.1.
Let S be a directed set and ( X s ) s ∈ S be a net of subsets of X . We saythat ( X s ) s ∈ S is asymptotically compact if for every directed set I , every subnet ( X s i ) i ∈ I of ( X s ) s ∈ S , and every ( y i ) i ∈ I ∈ Q i ∈ I X s i , the net ( y i ) i ∈ I in X has a convergent subnet. Remark . If there exists s ∈ S such that X s = ∅ for all s ≥ s , then it holds that( X s ) s ∈ S is asymptotically compact. 15 efinition 3.3. Let S be a directed set and ( X s ) s ∈ S be a net of subsets of X . We saythat ( X s ) s ∈ S is weakly asymptotically compact if for every directed set I , every monotonefinal map h : I → S , and every ( y i ) i ∈ I ∈ Q i ∈ I S t ∈ S,t ≥ h ( i ) X t , the net ( y i ) i ∈ I in X has aconvergent subnet.By definition, the asymptotic compactness implies the weak asymptotic compactness.It is not apparent whether the converse holds or not. We note that every limit y of aconvergent subnet of ( y i ) i ∈ I in Definitions 3.1 and 3.3 necessarily belongs to L ( X s ) s ∈ S from Theorem 2.15. Remark . Suppose that ( X s ) s ∈ S is weakly asymptotically compact. Then for every( x s ) s ∈ S ∈ Q s ∈ S X s , the net ( x s ) s ∈ S in X has a convergent subnet. Therefore, L ( X s ) s ∈ S is nonempty if every X s is nonempty. Lemma 3.5.
Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S is weakly asymptotically compact, then ( X s ) s ∈ S converges from above to L ( X s ) s ∈ S .Proof. We suppose the contrary and derive a contradiction. Then there is an open neigh-borhood U of L ( X s ) s ∈ S such that for all s ∈ S , S t ∈ S,t ≥ s X t ∩ ( X \ U ) = ∅ holds. Therefore,we can choose a net ( y s ) s ∈ S in X so that y s ∈ [ t ∈ S,t ≥ s X t ∩ ( X \ U )for all s ∈ S . By the weak asymptotic compactness, the net ( y s ) s ∈ S has a convergentsubnet. From Lemma 2.12 and Theorem 2.15, we have L ( X s ) s ∈ S ∩ ( X \ U ) = ∅ . This is acontradiction. Remark . The following properties are equivalent from Theorem 2.47:(a) X is compact.(b) For every directed set S = ( S, ≤ ) and every net ( X s ) s ∈ S of nonempty subsets of X ,( X s ) s ∈ S is asymptotically compact.(c) For every directed set S = ( S, ≤ ) and every net ( X s ) s ∈ S of nonempty subsets of X ,( X s ) s ∈ S is weakly asymptotically compact.We next study the asymptotic compactness when X is locally compact. Here X is saidto be locally compact if every point in X has a compact neighborhood. It is straightforwardto show that every compact set has a compact neighborhood when X is locally compact. Lemma 3.7.
Suppose that X is locally compact. Then for every directed set A = ( A, ≤ ) and every net ( x α ) α ∈ A in X converging from above to some nonempty compact set K ⊂ X , ( x α ) α ∈ A has a convergent subnet.Proof. We can choose a compact neighborhood U of K . The assumption implies that thereis α ∈ A such that for all α ∈ A , α ≥ α implies x α ∈ U . Let A := { α ∈ A : α ≥ α } bea directed subset of A . Then ( x α ) α ∈ A has a convergent subnet by the compactness of U from Theorem 2.47. This shows that ( x α ) α ∈ A also has a convergent subnet. Lemma 3.8.
Suppose that X is locally compact. Let S be a directed set and ( X s ) s ∈ S bea net of subsets of X . If ( X s ) s ∈ S converges from above to some compact set K ⊂ X , then ( X s ) s ∈ S is asymptotically compact. roof. Let I be a directed set and ( X s i ) i ∈ I be a subnet of ( X s ) s ∈ S . We only have toconsider the case that K and Q i ∈ I X s i are nonempty. Let ( y i ) i ∈ I ∈ Q i ∈ I X s i . Since( X s ) s ∈ S converges from above to K , the net ( y i ) i ∈ I in X converges from above to K .Therefore, ( y i ) i ∈ I has a convergent subnet from Lemma 3.7. This shows that ( X s ) s ∈ S isasymptotically compact. In this paper, we use the following terminologies.
Definition 3.9.
Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of subsets of X . We say that ( X s ) s ∈ S is limit set compact if (i) the limit set L ( X s ) s ∈ S is a nonemptycompact set and (ii) ( X s ) s ∈ S converges from above to L ( X s ) s ∈ S . Definition 3.10.
Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of subsets of X . We say that ( X s ) s ∈ S is eventually Lagrange stable if there exists s ∈ S such that S t ∈ S,t ≥ s X t is relatively compact, i.e., the closure is compact. Remark . The terminology of positive Lagrange stability has been used for a flow Φ : R + × X → X in a topological space X as follows (e.g., see [4]): The motion Φ ( t, x )is said to be positively Lagrange stable if the positive orbit { Φ ( t, x ) : t ∈ R + } is relativelycompact.The following theorem is considered to be a generalization of Theorem 2.47. Corollary 3.12.
The following properties are equivalent: (a) X is compact. (b) For every directed set S = ( S, ≤ ) and every net ( X s ) s ∈ S of nonempty subset of X , ( X s ) s ∈ S is limit set compact. (c) For every directed set S = ( S, ≤ ) and every net ( X s ) s ∈ S of nonempty subset of X , L ( X s ) s ∈ S = ∅ .Proof. (a) ⇒ (b): The non-emptiness of L ( X s ) s ∈ S follows by Lemma 2.46 because the net( Y s ) s ∈ S of subsets of X given by Y s := cl [ t ∈ S,t ≥ s X t ( s ∈ S )is a family of closed sets of X with the finite intersection property. The compactness of L ( X s ) s ∈ S also follows by the compactness of X . The convergence from above of ( X s ) s ∈ S to L ( X s ) s ∈ S is a consequence of Lemma 3.5.(b) ⇒ (c): This is obvious.(c) ⇒ (a): This holds from Theorem 2.47. Lemma 3.13.
Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of nonempty subsetsof X . If ( X s ) s ∈ S is eventually Lagrange stable, then the following statements hold:
1. ( X s ) s ∈ S converges to some nonempty closed compact set.
2. ( X s ) s ∈ S is asymptotically compact. Furthermore, ( X s ) s ∈ S is limit set compact. roof. By the eventual Lagrange stability, there is s ∈ S such that K := cl S t ∈ S,t ≥ s X t is a nonempty closed compact set.1. Since X s ⊂ K holds for all s ≥ s , it holds that ( X s ) s ∈ S converges to K .2. Let I = ( I, ≤ ) be a directed set and ( X s i ) i ∈ I be a subnet of ( X s ) s ∈ S . We onlyhave to consider the case that Q i ∈ I X s i is nonempty. Let ( y i ) i ∈ I ∈ Q i ∈ I X s i . For theabove s , there is i ∈ I such that s i ≥ s holds for all i ≥ i . Let I := { i ∈ I : i ≥ i } be a directed subset of I . Then ( y i ) i ∈ I becomes a net in K . Therefore, this net has aconvergent subnet from Theorem 2.47. This implies that ( y i ) i ∈ I also has a convergentsubnet. Thus, ( X s ) s ∈ S is asymptotically compact. From Lemma 3.5, ( X s ) s ∈ S convergesfrom above to the nonempty closed set L ( X s ) s ∈ S . Furthermore, L ( X s ) s ∈ S is compactbecause L ( X s ) s ∈ S ⊂ K . Therefore, the limit set compactness follows.We finally study a connection between the eventual Lagrange stability and the limitset compactness in locally compact regular spaces. The following fact is crucial. Lemma 3.14 (ref. [8]) . Suppose that X is a regular space. Then for every compact set A ⊂ X , cl( A ) is also compact. Lemma 3.15.
Suppose that X is a locally compact regular space. Let S = ( S, ≤ ) be adirected set and ( X s ) s ∈ S be a net of nonempty subsets of X . Then the following propertiesare equivalent: (a) ( X s ) s ∈ S converges from above to some nonempty compact set. (b) ( X s ) s ∈ S is eventually Lagrange stable. (c) ( X s ) s ∈ S is asymptotically compact and limit set compact.Proof. (a) ⇒ (b): Let K ⊂ X be a nonempty compact set to which ( X s ) s ∈ S convergesfrom above. We can choose a compact neighborhood U of K and s ∈ S such that S t ∈ S,t ≥ s X t ⊂ U . This shows cl S t ∈ S,t ≥ s X t ⊂ cl( U ), where cl( U ) is also compact fromLemma 3.14. Therefore, ( X s ) s ∈ S is eventually Lagrange stable.(b) ⇒ (c): This follows by Lemma 3.13.(c) ⇒ (a): This is obvious.The following is a consequence of Lemma 3.15 because every locally compact Hausdorffspace is regular (ref. [8]). The proof can be omitted. Corollary 3.16.
Suppose that X is a locally compact Hausdorff space. Let S = ( S, ≤ ) be a directed set and ( X s ) s ∈ S be a net of nonempty subsets of X . Then the followingproperties are equivalent: (a) ( X s ) s ∈ S converges from above to some nonempty compact set. (b) ( X s ) s ∈ S is eventually Lagrange stable. (c) ( X s ) s ∈ S is asymptotically compact and limit set compact. Let X be a set. In this section, we will prove that the asymptotic compactness and thelimit set compactness are equivalent for any net of nonempty subsets in uniformizablespaces. 18 .1 Uniformity and uniform spaces For the purpose stated above, we recall the definition of uniform spaces.
Notation . Let X be a set. • Let ∆ X denote the diagonal set { ( x, x ) ∈ X × X : x ∈ X } . • For every subset U ⊂ X × X , let U − := { ( x, y ) ∈ X × X : ( y, x ) ∈ U } . • For every subsets
U, V ⊂ X × X , let V ◦ U := { ( x, y ) ∈ X × X : ( x, z ) ∈ U and ( z, y ) ∈ V for some z ∈ X } . Definition 4.1 (ref. [8]) . Let X be a set. A nonempty collection U of subsets of X × X is called a uniformity if the following properties are satisfied:(U1) Every U ∈ U contains the diagonal ∆ X .(U2) For every U ∈ U , the inverse U − also belongs to U .(U3) For every U ∈ U , there exists V ∈ U such that the composition V ◦ V is a subset of U .(U4) For every U, V ∈ U , U ∩ V ∈ U .(U5) For every U ∈ U and every subset V ⊂ X × X containing U , V ∈ U holds. U ∈ U is said to be symmetric if U − = U . In (U3), one can assume that V is symmetricby considering V ∩ V − . The pair ( X, U ) is called a uniform space .We note that a uniformity U on X is considered as a directed poset with the partialorder ≤ defined as follows: U ≤ U if U ⊃ U . Notation . Let X be a set. For every subset U ⊂ X × X , every x ∈ X , and every subset E ⊂ X , let U [ x ] := { y ∈ X : ( x, y ) ∈ U } , U [ E ] := [ x ∈ E U [ x ] . We note that U [ ∅ ] is interpreted as ∅ .A topological space X is said to be uniformizable if the topology is the uniform topology T of some uniformity U defined as follows: T ∈ T if for every x ∈ T , there exists U ∈ U such that U [ x ] ⊂ T . In this subsection, we prove that the convergence from above to some nonempty compactset implies the asymptotic compactness. 19 emma 4.2.
Suppose that X = ( X, U ) is a uniform space. Let S = ( S, ≤ ) be a directedset, ( x s ) s ∈ S be a net in X , and K ⊂ X be a nonempty subset. Let I = ( I, ≤ ) be the subsetof the product directed set S × U defined by I = { ( s, U ) ∈ S × U : x s ∈ U [ K ] } . If ( x s ) s ∈ S converges from above to K , then for every ( s , U ) , ( s , U ) ∈ S × U , there existsan upper bound ( s, U ) ∈ I of ( s , U ) , ( s , U ) .Proof. We note that I is nonempty by the assumption. Let ( s , U ) , ( s , U ) ∈ S × U begiven. We choose U := U ∩ U ∈ U . Since U [ K ] is a neighborhood of K , there is s ∈ S such that for all s ∈ S , s ≥ s implies x s ∈ U [ K ]. Therefore, by choosing an upper bound s ∈ S of s , s , s , we have an upper bound ( s, U ) ∈ I of ( s , U ) , ( s , U ). Remark . In particular, the following hold:1. I = ( I, ≤ ) is a directed set.2. I is a cofinal subset of S × U .The following theorem is an extension of Theorem 2.47. Theorem 4.4.
Suppose that X is a uniformizable space. Then for every directed set S = ( S, ≤ ) and every net ( x s ) s ∈ S converging from above to some nonempty compact set K ⊂ X , ( x s ) s ∈ S has a subnet converging to some point in K .Proof. Step 1.
We choose a uniformity U so that the topology of X is the uniformtopology of U . Let I be the directed set given in Lemma 4.2. Then for every ( s, U ) ∈ I ,there is y s,U ∈ K such that x s ∈ U [ y s,U ], i.e.,( y s,U , x s ) ∈ U. From Theorems 2.47 and 2.10, there are a directed set J = ( J, ≤ ) and a monotone finalmap h : J → I such that ( y h ( j ) ) j ∈ J converges to some point y ∈ K . Let( s j , U j ) := h ( j )for each j ∈ J . Since I is a cofinal subset of S × U , the maps J ∋ j s j ∈ S, J ∋ j U j ∈ U are also monotone and final. Step 2.
We claim that the subnet ( x s j ) j ∈ J of ( x s ) s ∈ S converges to y . Let U ∈ U begiven. We choose a symmetric U ′ ∈ U so that U ′ ◦ U ′ ⊂ U . Then there is j ∈ J such that j ≥ j implies U j ≥ U ′ . Since ( y h ( j ) ) j ∈ J converges to y , there is j ∈ J such that j ≥ j implies y h ( j ) ∈ U ′ [ y ] . Let j ∈ J be an upper bound of j , j . Then for all j ≥ j , we have ( y h ( j ) , x s j ) ∈ U j ⊂ U ′ and ( y, y h ( j ) ) ∈ U ′ , which shows ( y, x s j ) ∈ U, i.e., x s j ∈ U [ y ]. Therefore, ( x s j ) j ∈ J converges to y .20s a corollary, we can obtain the asymptotic compactness from the convergence fromabove to some compact set. Corollary 4.5.
Suppose that X is a uniformizable space. Let S be a directed set and ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S converges from above to some compact set K ⊂ X , then ( X s ) s ∈ S is asymptotically compact.Proof. Let I be a directed set and ( X s i ) i ∈ I be a subnet of ( X s ) s ∈ S . We only have toconsider the case that K and Q i ∈ I X s i are nonempty. Let ( y i ) i ∈ I ∈ Q i ∈ I X s i be given.The convergence of ( X s ) s ∈ S from above to K implies that the net ( y i ) i ∈ I in X convergesfrom above to K . Therefore, ( y i ) i ∈ I has a convergent subnet by applying Theorem 4.4.This shows that ( X s ) s ∈ S is asymptotically compact. In the following theorem, we will prove that the weak asymptotic compactness inducesthe limit set compactness.
Theorem 4.6.
Suppose that X is a uniformizable space. Let S = ( S, ≤ ) be a directed setand ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S is weakly asymptotically compact, then ( X s ) s ∈ S is compact. Consequently, furthermore, if every X s is nonempty, then ( X s ) s ∈ S islimit set compact.Proof. Let A = ( A, ≤ ) be a directed set and ( z α ) α ∈ A be a net in L ( X s ) s ∈ S . We will showthat ( z α ) α ∈ A has a subnet converging to some point in L ( X s ) s ∈ S . Then the compactnessof L ( X s ) s ∈ S follows by Theorem 2.47. Step 1.
We choose a uniformity U so that the topology of X is the uniform topologyof U . Let B := S × U × A be the product directed set. By the definition of L ( X s ) s ∈ S , U [ z α ] ∩ S t ∈ S,t ≥ s X t = ∅ holdsfor every ( s, U, α ) ∈ B . Therefore, we can choose a net ( y s,U,α ) ( s,U,α ) ∈ B in X so that y s,U,α ∈ U [ z α ] ∩ [ t ∈ S,t ≥ s X t for all ( s, U, α ) ∈ B . Let h : B → S be the monotone final map defined by h ( s, U, α ) = s (( s, U, α ) ∈ B ) . Since y s,U,α ∈ [ t ∈ S,t ≥ h ( s,U,α ) X t for all ( s, U, α ) ∈ B , the net ( y s,U,α ) ( s,U,α ) ∈ B has a subnet converging to some y ∈ L ( X s ) s ∈ S by the weak asymptotic compactness of ( X s ) s ∈ S . From Theorem 2.10, there are a directedset C = ( C, ≤ ) and a monotone final map g : C → B such that the net ( y g ( γ ) ) γ ∈ C convergesto y . Step 2.
Let ( s γ , U γ , α γ ) := g ( γ ) . γ ∈ C . Then the maps C ∋ γ s γ ∈ S, C ∋ γ U γ ∈ U , C ∋ γ α γ ∈ A are also monotone and final. We claim that the subnet ( z α γ ) γ ∈ C of ( z α ) α ∈ A converges to y . Let U ∈ U be given. We choose a symmetric U ′ ∈ U so that U ′ ◦ U ′ ⊂ U . Then thereexists γ ∈ C such that for all γ ≥ γ , U γ ≥ U ′ , y g ( γ ) ∈ U ′ [ y ] , which shows ( z α γ , y g ( γ ) ) ∈ U γ ⊂ U ′ . Therefore, for all γ ≥ γ , we have( y, z α γ ) ∈ U. This shows that ( z α γ ) γ ∈ C converges to y . Finally, the limit set compactness of ( X s ) s ∈ S under the non-emptiness of every X s is a consequence of Lemma 3.5.We obtain the equivalence of the asymptotic compactness and the limit set compactnessin uniformizable spaces by combining the results of this section. Theorem 4.7.
Suppose that X is a uniformizable space. Let S be a directed set and ( X s ) s ∈ S be a net of nonempty subsets of X . Then the following properties are equivalent: (a) ( X s ) s ∈ S converges from above to some nonempty compact set. (b) ( X s ) s ∈ S is asymptotically compact. (c) ( X s ) s ∈ S is weakly asymptotically compact. (d) ( X s ) s ∈ S is limit set compact.Proof. (a) ⇒ (b): This is Corollary 4.5. (b) ⇒ (c): This is obvious by definition. (c) ⇒ (d): This is Theorem 4.6. (d) ⇒ (a): This is obvious.This is an extension of Lemma 3.15 because every locally compact regular space iscompletely regular, which is equivalent to the uniformizability (ref. [8]). Throughout this section, let X be a topological space. The purpose of this section is toinvestigate the sequential versions of the asymptotic compactness and the weak asymptoticcompactness introduced in Section 3. For the purpose stated above, we introduce the notion of the sequentiality of directed setsas follows.
Definition 5.1.
We say that a directed set S is sequential if there exists a sequence( s n ) ∞ n =1 in S such that s n → bd( S ) as n → ∞ .22s shown in the following lemma, we can choose a monotone final sequence in everysequential directed set. Therefore, we may assume that the sequence ( s n ) ∞ n =1 in Defini-tion 5.1 is monotone and final. Lemma 5.2.
Let S = ( S, ≤ ) be a directed set and ( t n ) ∞ n =1 be a sequence in S with t n → bd( S ) as n → ∞ . Then there exists a monotone final sequence ( s n ) ∞ n =1 in S such that s n ≥ t n for all n ≥ .Proof. We construct a sequence ( s n ) ∞ n =1 satisfying s n +1 ≥ s n and s n ≥ t n for all n ≥ s = t .(ii) Let n ≥ s , . . . , s n ∈ S are chosen so that s i ≥ t i for all1 ≤ i ≤ n and s ≤ · · · ≤ s n . Since S is directed, we can choose an upper bound s n +1 ∈ S of s n , t n +1 . Then s n +1 satisfies s n +1 ≥ s n and s n +1 ≥ t n +1 .Since t n → bd( S ) as n → ∞ , the sequence ( s n ) ∞ n =1 is final. In this subsection, we introduce the following sequential limit sets.
Definition 5.3.
Let S be a sequential directed set and ( X s ) s ∈ S be a net of subsets of X .We define a subset L seq ( X s ) s ∈ S of X as follows: y ∈ L seq ( X s ) s ∈ S if there exist a subnetof ( X s n ) ∞ n =1 of ( X s ) s ∈ S and ( y n ) ∞ n =1 ∈ Q ∞ n =1 X s n such that the sequence ( y n ) ∞ n =1 in X converges to y . We call L seq ( X s ) s ∈ S the sequential limit set of ( X s ) s ∈ S .We note that L seq ( X s ) s ∈ S ⊂ L ( X s ) s ∈ S holds from Theorem 2.15. We now show thatthe sequential limit set is identical to the limit set when X is first-countable and S issequential. We recall that X is said to be first-countable if every point in X has a countablelocal base. Remark . Let x ∈ X be given. Suppose that { U n ∈ N x : n ≥ } is a countable localbase at x . By considering a family { U ′ n : n ≥ } in N x given by U ′ n := U ∩ · · · ∩ U n ( n ≥ , we may assume that U n ⊃ U n +1 holds for all n ≥
1, i.e., ( U n ) ∞ n =1 is a monotone finalsequence in N x .The following lemma shows that the concept of subsequences is suffice for first-countablespaces. Lemma 5.5 (ref. [8]) . Let x ∈ X be a cluster point of some sequence ( x n ) ∞ n =1 in X . If X is first-countable, then ( x n ) ∞ n =1 has a subsequence converging to x . Lemma 5.6.
Let S = ( S, ≤ ) be a sequential directed set and ( X s ) s ∈ S be a net of subsetsof X . If X is first-countable, then L ( X s ) s ∈ S = L seq ( X s ) s ∈ S holds. roof. We only have to consider the case L ( X s ) s ∈ S = ∅ . Let y ∈ L ( X s ) s ∈ S be given. Wechoose a monotone final sequence ( s n ) ∞ n =1 in S and a monotone final sequence ( U n ) ∞ n =1 in N y . Then for every n ≥ U n ∩ S t ∈ S,t ≥ s n X t = ∅ holds. Therefore, we can choose asequence ( y n ) ∞ n =1 in X so that y n ∈ U n ∩ [ t ∈ S,t ≥ s n X t for all n ≥
1. This implies that ( y n ) ∞ n =1 converges to y . The above also implies theexistence of a sequence ( t n ) ∞ n =1 in S such that t n ≥ s n and y n ∈ X t n for all n ≥ y ∈ L seq ( X s ) s ∈ S holds. In this subsection, we introduce the sequential versions of the asymptotic compactnessand the weak asymptotic compactness.
Definition 5.7 (cf. [11]) . Let S be a directed set and ( X s ) s ∈ S be a net of subsets of X .We say that ( X s ) s ∈ S is asymptotically sequentially compact if (i) S is sequential and (ii) forevery subnet ( X s n ) ∞ n =1 of ( X s ) s ∈ S and every ( y n ) ∞ n =1 ∈ Q ∞ n =1 X s n , the sequence ( y n ) ∞ n =1 in X has a convergent subsequence. Remark . Suppose that S is sequential. If there exists s ∈ S such that X s = ∅ for all s ≥ s , then it holds that ( X s ) s ∈ S is asymptotically sequentially compact. Remark . Let X = ( X, d ) be a metric space, Φ : R + × X → X be a semiflow, and E ⊂ X be a subset. In [11], Φ is said to be asymptotically compact on E if for every sequence( t n ) ∞ n =1 in R + with t n → ∞ as n → ∞ and every sequence ( x n ) ∞ n =1 in E , ( Φ ( t n , x n )) ∞ n =1 has a convergent subsequence. The asymptotic sequential compactness generalizes thisnotion. Definition 5.10.
Let S be a directed set and ( X s ) s ∈ S be a net of subsets of X . We saythat ( X s ) s ∈ S is weakly asymptotically sequentially compact if (i) S is sequential and (ii)for every monotone final sequence ( s n ) ∞ n =1 in S and every ( y n ) ∞ n =1 ∈ Q ∞ n =1 S t ∈ S,t ≥ s n X t ,the sequence ( y n ) ∞ n =1 in X has a convergent subsequence. Remark . Suppose that S is sequential, ( s n ) ∞ n =1 is a monotone final sequence in S ,and ( X s ) s ∈ S is weakly asymptotically sequentially compact. Then for every ( y n ) ∞ n =1 ∈ Q ∞ n =1 X s n , the sequence ( y n ) ∞ n =1 in X has a convergent subsequence. Therefore, L seq ( X s ) s ∈ S is nonempty if every X s is nonempty.By definition, the asymptotic sequential compactness implies the weak asymptotic se-quential compactness. It is not apparent whether the converse holds or not. The followingis the sequential version of Lemma 3.5. Lemma 5.12.
Let S = ( S, ≤ ) be a sequential directed set and ( X s ) s ∈ S be a net of subsetsof X . If ( X s ) s ∈ S is weakly asymptotically sequentially compact, then ( X s ) s ∈ S convergesfrom above to L ( X s ) s ∈ S .Proof. We suppose that ( X s ) s ∈ S does not converge from above to L ( X s ) s ∈ S . Then thereexists an open neighborhood U of L ( X s ) s ∈ S such that for all s ∈ S , [ t ∈ S,t ≥ s X t ∩ ( X \ U ) = ∅ . y s ) s ∈ S in X so that y s ∈ [ t ∈ S,t ≥ s X t ∩ ( X \ U )for all s ∈ S . By choosing a monotone final sequence ( s n ) ∞ n =1 in S , we can obtain ∅ 6 = L seq ( X s ) s ∈ S ∩ ( X \ U ) ⊂ L ( X s ) s ∈ S ∩ ( X \ U )in the similar way to Lemma 3.5. This is a contradiction.In the remainder of this subsection, we study a relation between the convergence fromabove to a compact set and the asymptotic sequential compactness when S is sequentialand X is first-countable. For this purpose, we use the following lemma. Lemma 5.13.
Let ( x n ) ∞ n =1 be a sequence in X . If ( x n ) ∞ n =1 converges from above tosome nonempty compact set K ⊂ X , then it has a convergent subnet. Consequently,furthermore, if X is first-countable, then ( x n ) ∞ n =1 has a convergent subsequence. We give the proof for the sake of completeness although it is standard.
Proof of Lemma 5.13.
Let ¯ K := { x n : n ≥ } ∪ K. We claim that ¯ K is compact. Then the existence of a convergent subnet follows byTheorem 2.47. Let U be an open cover of ¯ K . Since U is also an open cover of K , there isa finite sub-collection U ⊂ U such that U is a cover of K . Then S U := S U ∈U U is aneighborhood of K . Therefore, there is n ≥ n ≥ n , x n ∈ [ U holds. Since { x n : 1 ≤ n < n } is a finite set, there is a finite sub-collection U ⊂ U such that U is a cover of { x n : 1 ≤ n < n } . This shows that U ∪ U is a subcover of¯ K . Therefore, the compactness of ¯ K is obtained. Finally, the existence of a convergentsubsequence under the first-countability of X is a consequence of Lemma 5.5.The following is a corollary of Lemma 5.13. Corollary 5.14.
Suppose that X is first-countable. Let S be a sequential directed set and ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S converges from above to some compact set K ⊂ X , then ( X s ) s ∈ S is asymptotically sequentially compact.Proof. Let ( X s n ) ∞ n =1 be a subnet of ( X s ) s ∈ S . We only have to consider the case that K and Q ∞ n =1 X s n are nonempty. Let ( y n ) ∞ n =1 ∈ Q ∞ n =1 X s n be given. By the assumption, thesequence ( y n ) ∞ n =1 in X converges from above to some nonempty compact set. Therefore,( y n ) ∞ n =1 has a convergent subsequence from Lemma 5.13. This shows that ( X s ) s ∈ S isasymptotically sequentially compact.Lemma 5.13 also gives the following statement. It should be compared with Lemma 5.12. Lemma 5.15.
Let S = ( S, ≤ ) be a sequential directed set and ( X s ) s ∈ S be a net of nonemptysubsets of X . If ( X s ) s ∈ S converges from above to some nonempty compact set, then thefollowing statements hold: L ( X s ) s ∈ S = ∅ .
2. ( X s ) s ∈ S converges from above to L ( X s ) s ∈ S .Proof.
1. We choose a monotone final sequence ( s n ) ∞ n =1 in S and ( x s ) s ∈ S ∈ Q s ∈ S X s . Then( x s n ) ∞ n =1 is a subnet of ( x s ) s ∈ S and converges from above to some nonempty compact set.From Lemma 5.13, ( x s n ) ∞ n =1 has a convergent subnet. Therefore, L ( X s ) s ∈ S is nonemptyfrom Theorem 2.15.2. We suppose that ( X s ) s ∈ S does not converge to L ( X s ) s ∈ S . Then there exists anopen neighborhood U of L ( X s ) s ∈ S such that for all s ∈ S , S t ∈ S,t ≥ s X t ∩ ( X \ U ) = ∅ holds.Therefore, we can choose a net ( y s ) s ∈ S in X so that y s ∈ [ t ∈ S,t ≥ s X t ∩ ( X \ U )for all s ∈ S . By choosing a monotone final sequence ( s n ) ∞ n =1 in S , we can obtain L ( X s ) s ∈ S ∩ ( X \ U ) = ∅ in the same way as the proof of the statement 1. This is a contradiction.The following is a corollary of Lemma 5.15. Corollary 5.16 (cf. [9]) . Suppose that X is Hausdorff or regular. Let S = ( S, ≤ ) be asequential directed set and ( X s ) s ∈ S be a net of nonempty subsets of X . Then ( X s ) s ∈ S converges from above to some nonempty compact set if and only if ( X s ) s ∈ S is limit setcompact.Proof. We need to show the only-if-part. Lemma 5.15 implies that L ( X s ) s ∈ S is nonemptyand ( X s ) s ∈ S converges from above to L ( X s ) s ∈ S . Let K be a nonempty compact set towhich ( X s ) s ∈ S converges from above. Then we have L ( X s ) s ∈ S ⊂ cl( K )from Lemma 2.50 or Lemma 2.52. Since cl( K ) is compact from Lemma 3.14, this impliesthe compactness of L ( X s ) s ∈ S by its closedness. Therefore, ( X s ) s ∈ S is limit set compact. Remark . Suppose that X is a regular Hausdorff space. Let Φ : R + × X → X be asemiflow and E ⊂ X be a subset. In [9, Theorem 2.10], it is proved that if E is attractedby some nonempty compact set under Φ , then the omega limit set ω Φ ( E ) of E for Φ is anonempty compact set which attracts E under Φ . As the following theorem shows, in Theorem 4.6, we can replace the weak asymptotic com-pactness with the weak asymptotic sequential compactness when X is pseudo-metrizableand S is sequential. 26 heorem 5.18 (cf. [11]) . Suppose that X is pseudo-metrizable. Let S = ( S, ≤ ) be asequential directed set and ( X s ) s ∈ S be a net of subsets of X . If ( X s ) s ∈ S is weakly asymp-totically sequentially compact, then L ( X s ) s ∈ S is compact. Consequently, furthermore, ifevery X s is nonempty, then ( X s ) s ∈ S is limit set compact. We give the proof for the sake of completeness although it is similar to the proof ofthe compactness of the omega limit set ω Φ ( E ) of E for a semiflow Φ : R + × X → X givenin [11]. Proof of Theorem 5.18.
Let d be a pseudo-metric generating the topology of X . Let( s n ) ∞ n =1 be a monotone final sequence in S and ( z n ) ∞ n =1 be a sequence in L ( X s ) s ∈ S . Thenfor every n ≥ B d ( z n ; 1 /n ) ∩ S t ∈ S,t ≥ s n X t = ∅ . Therefore, we can choose a sequence( y n ) ∞ n =1 in X so that y n ∈ B d (cid:18) z n ; 1 n (cid:19) ∩ [ t ∈ S,t ≥ s n X t for all n ≥
1. The sequence ( y n ) ∞ n =1 has a subsequence ( y n k ) ∞ k =1 converging to some y ∈ L ( X s ) s ∈ S by the weak asymptotic sequential compactness of ( X s ) s ∈ S . Then ( z n k ) ∞ k =1 also converges to y because d ( z n k , y ) ≤ d ( z n k , y n k ) + d ( y n k , y )holds for all k ≥
1. Therefore, it holds that L ( X s ) s ∈ S is sequentially compact. Thelimit set compactness of ( X s ) s ∈ S under the non-emptiness of every X s is a consequence ofLemma 5.12.We finally obtain the equivalence of the asymptotic sequential compactness and thelimit set compactness in pseudo-metrizable spaces when S is sequential. Theorem 5.19.
Suppose that X is pseudo-metrizable. Let S be a sequential directedset and ( X s ) s ∈ S be a net of nonempty subsets of X . Then the following properties areequivalent: (a) ( X s ) s ∈ S converges from above to some nonempty compact set. (b) ( X s ) s ∈ S is asymptotically sequentially compact. (c) ( X s ) s ∈ S is weakly asymptotically sequentially compact. (d) ( X s ) s ∈ S is limit set compact.Proof. (a) ⇒ (b): This follows by Corollary 5.14 because every pseudo-metrizable spaceis first-countable. (b) ⇒ (c): This follows by definition. (c) ⇒ (d): This follows byTheorem 5.18. (d) ⇒ (a): This is obvious.The following corollary is a combination of Theorems 4.7 and 5.19. The proof can beomitted. Corollary 5.20.
Suppose that X is pseudo-metrizable. Let S be a sequential directed setand ( X s ) s ∈ S be a net of nonempty subsets of X . Then ( X s ) s ∈ S is asymptotically compactif and only if it is asymptotically sequentially compact. cknowledgments This work was supported by the Research Alliance Center for Mathematical Sciencesof Tohoku University, the Advanced Institute for Materials Research of Tohoku Uni-versity, the Research Institute for Mathematical Sciences for an International Joint Us-age/Research Center located in Kyoto University, Grant-in-Aid for JSPS Fellows GrantNumber JP15J02604, JSPS Grant-in-Aid for Scientific Research on Innovative AreasGrant Number JP17H06460, and JSPS Grant-in-Aid for Young Scientists Grant Num-ber JP19K14565.
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