aa r X i v : . [ m a t h . GN ] J u l ON QUASI κ -METRIZABLE SPACES VESKO VALOV
Abstract.
The aim of this paper is to investigate the class ofquasi κ -metrizable spaces. This class is invariant with respect toarbitrary products and contains Shchepin’s [8] κ -metrizable spacesas a proper subclass. Introduction
Recall that a κ -metric [8] on a space X is a non-negative function ρ ( x, C ) of two variables, a point x ∈ X and a regularly closed set C ⊂ X , satisfying the following conditions:K1) ρ ( x, C ) = 0 iff x ∈ C ;K2) If C ⊂ C ′ , then ρ ( x, C ′ ) ≤ ρ ( x, C ) for every x ∈ X ;K3) ρ ( x, C ) is continuous function of x for every C ;K4) ρ ( x, S C α ) = inf α ρ ( x, C α ) for every increasing transfinite fam-ily { C α } of regularly closed sets in X .A κ -metric on X is said to be regular if it satisfy also next conditionK5) ρ ( x, C ) ≤ ρ ( x, C ′ ) + ρ ( C ′ , C ) for any x ∈ X and any two regu-larly closed sets C, C ′ in X , where ρ ( C ′ , C ) = sup { ρ ( y, C ) : y ∈ C ′ } .We say that a function ρ ( x, C ) is an quasi κ -metric (resp., a regularquasi κ -metric) on X if it satisfies the axioms K − K
4) (resp., K − K ∗ For any C there is a dense open subset V of X \ C such that ρ ( x, C ) > x ∈ V .Obviously, we can assume that ρ ( x, C ) ≤ x and C , in such acase we say that ρ is a normed quasi κ -metric.Quasi κ -metrizable spaces were introduced in [9]. Our interest of thisclass was originated by Theorem 1.4 from [9] stating that a compactspace is quasi κ -metrizable if and only if it is skeletally generated. Mathematics Subject Classification.
Primary 54C10; Secondary 54F65.
Key words and phrases. compact spaces, I-favorable spaces, κ -metrizable spaces,skeletal maps, skeletally generated spaces.Research supported in part by NSERC Grant 261914-13. Unfortunately, the presented there proof of the implication that anyskeletally generated compactum is quasi κ -metrizable is not correct.Despite of this incorrectness, the class of quasi κ -metrizable is veryinteresting. It is closed with respect to arbitrary products and containsas a proper subclass the κ -metrizable spaces. The aim of this paper isto investigate the class of quasi κ -metrizable spaces, and to provide acorrect characterization of skeletally generated spaces.The class of skeletally generated spaces was introduced in [10]. Ac-cording to [9, Theorem 1.1], a space is skeletally generated iff it is I -favorable in the sense of [2]. Recall that a map f : X → Y is skeletalif Int f ( U ) = ∅ ) for every open U ⊂ X . A space X is skeletally gen-erated [10] if there is an inverse system S = { X α , p βα , A } of separablemetric spaces X α and skeletal surjective bounding maps p βα satisfy-ing the following conditions: (1) the index set A is σ -complete (everycountable chain in A has a supremum in A ); (2) for every countablechain { α n } n ≥ ⊂ A with β = sup { α n } n ≥ the space X β is a (dense) sub-set of lim ← { X α n , p α n +1 α n } ; (3) X is embedded in lim ← S and p α ( X ) = X α for each α , where p α : lim ← S → X α is the α -th limit projection. If inthe above definition all bounding maps p βα are open, we say that X isopenly generated.All topological spaces are Tychonoff and the single-valued maps arecontinuous. The paper is organized as follows: Section 2 contains theproof that any product of quasi κ -metrizable spaces is also quasi κ -metrizable. In Section 3 we provide some additional properties of quasi κ -metrizable spaces. For example, it is shown that this property is pre-served by open and perfect surjections, and that the ˇCech-Stone com-pactification of any pseudocompact quasi κ -metrizable space is quasi κ -metrizable. The results from Section 3 imply that there exist quasi κ -metrizable spaces which are not κ -metrizable. In Section 4 we intro-duce a similar wider class of spaces, the weakly κ -metrizable spaces,and proved that a compact space is skeletally generated if and only if itis weakly κ -metrizable. Hence, every skeletally generated space is alsoweakly κ -metrizable. The converse implication is interesting for spaceswith a countable cellularity only, but it is still unknown, see Questions4.3 - 4.4. 2. Products of quasi κ -metrizable spaces Let B be a base for a space X consisting of regularly open sets. Areal-valued function ξ : X × B → [0 ,
1] will be called a π -capacity if itsatisfies the following conditions: uasi κ -metrics 3 E1) ξ ( x, U ) = 0 for x U , and 0 ≤ ξ ( x, U ) ≤ x ∈ U .E2) For any U ∈ B the set { x ∈ U : ξ ( x, U ) > } is dense in U .E3) For any U the function ξ ( x, U ) is lower semi-continuous, i.e if ξ ( x , U ) > a for some x ∈ X and a ∈ R , then there is aneighborhood O x with ξ ( x, U ) > a for all x ∈ O x .E4) For any two mappings U : T → B and λ : T → X , where T is a set with an ultrafilter F , such that the limit e λ = lim F λ ( t )exists and lim F ξ ( λ ( t ) , U ( t )) > a >
0, then there exists e U ∈ B such that e U ⊂ lim F U ( t ) and ξ ( e λ, e U ) > a . Here, lim F U ( t ) = T F ∈F S t ∈ F U ( t ).A capacity is called regular if it satisfies also the following condition:E5) If ξ ( x, U ) > a >
0, there exists U a ∈ B such that ξ ( x, U a ) ≥ a and ξ ( y, U ) ≥ ξ ( x, U ) − a for all y ∈ U a .Our definition of a π -capacity is almost the same as the Shchepin’sdefinition [7] of capacity, the only difference is that Shchepin requires ξ ( x, U ) > x ∈ U . Lemma 2.1.
Suppose ξ : X × B → [0 , is a (regular) π -capacity on X . Then the function ρ ξ ( x, C ) , ρ ξ ( x, C ) = 0 if x ∈ C and ρ ξ ( x, C ) =sup { ξ ( x, U ) : U ∩ C = ∅ } otherwise, is a (regular) quasi κ -metric on X .Proof. Suppose ξ is a π -capacity on X . Clearly, ρ ξ satisfies condition K ρ ξ alsosatisfies conditions K − K K ∗ , let C be aproper regularly closed subset of X . Then there is a subfamily { U α : α ∈ A } of B covering X \ C . For every α the set V α = { x ∈ U α : ξ ( x, U α ) > } is dense in U α . So V = S α ∈ A V α is dense in X \ C and ρ ξ ( x, C ) > x ∈ V .Let show that if ξ is a regular π -capacity, then ρ ξ satisfies condition K C, C ′ are two regularly closed subsets of X and x ∈ X . Obviously, ρ ξ ( x, C ) ≤ ρ ξ ( x, C ′ ) implies ρ ξ ( x, C ) ≤ ρ ξ ( x, C ′ ) + ρ ξ ( C ′ , C ). So, let ρ ξ ( x, C ) > ρ ξ ( x, C ′ ), and choose an integer m suchthat ρ ξ ( x, C ) > ρ ξ ( x, C ′ ) + 1 /n for all n ≥ m . Hence, there is U ∈B such that U ∩ C = ∅ and ξ ( x, U ) > a n = ρ ξ ( x, C ′ ) + 1 /n . So,according to condition E U a n ∈ B such that ξ ( x, U a n ) ≥ a n and ξ ( x, U ) ≤ ξ ( y, U ) + a n for all y ∈ U a n . Since ξ ( x, U a n ) ≥ a n , U a n ∩ C ′ = ∅ (otherwise we would have ρ ξ ( x, C ′ ) ≥ ρ ξ ( x, C ′ ) + 1 /n ).Hence, ξ ( x, U ) ≤ ξ ( z, U ) + a n for every z ∈ U a n ∩ C ′ , which yields ξ ( x, U ) ≤ ρ ξ ( C ′ , C ) + ρ ξ ( x, C ′ ) + 1 /n for all n ≥ m and U ∈ B with U ∩ C = ∅ . Therefore, ρ ξ ( x, C ) ≤ ρ ξ ( x, C ′ ) + ρ ξ ( C ′ , C ). (cid:3) Lemma 2.2.
Let ρ be a (regular) normed quasi κ -metric on X and B be the family of all regularly open subsets of X . Then the formula ξ ( x, U ) = sup { ρ ( x, C ) : C ∪ U = X } defines a (regular) π -capacity on X .Proof. It is easy to show that ξ satisfies conditions E
1) and E E
4) was established in [7, Lemma 3] in the case ρ is a κ -metric, butthe same proof works for quasi κ -metrics as well. Let show condition E U ∈ B and consider the family B U = { G ∈ B : G U } .For every G ∈ B U the set V G = { x ∈ G : ρ ( x, C G ) > } is open andnon-empty, where C G = X \ G . Hence, V = S G ∈B U V G is dense in U .Moreover, ξ ( x, U ) ≥ ρ ( x, C G ) for every G ∈ B U and x ∈ V G because U ∪ C G = X . So, ξ ( x, U ) > x ∈ V .It remains to show that ξ is regular, i.e. it satisfies E ρ is regular. Let U ∈ B and ξ ( x, U ) > a > x ∈ U . Since C U = X \ U is the smallest regularly closed subset C of X with U ∪ C = X , we have ξ ( z, U ) = ρ ( z, C U ) for all z ∈ X . The set W = { y ∈ X : ρ ( y, C U ) > ρ ( x, C U ) − a } is open and non-empty because ρ ( · , C )is continuous and x ∈ W . Let U a ∈ B be the set U a = IntW and C U a = X \ U a . Observe that W ⊂ U , which implies U a ⊂ U and C U ⊂ C U a . Then, by K ρ ( x, C U ) ≤ ρ ( x, C U a ) + ρ ( C U a , C U ). Consequently, ρ ( x, C U a ) ≥ ρ ( x, C U ) − ρ ( C U a , C U ). On the other hand, ρ ( C U a , C U ) =sup y ∈ C Ua ρ ( y, C U ) = sup y ∈ C Ua \ C U ρ ( y, C U ) because ρ ( y, C U ) = 0 for all y ∈ C U . Observe also that ρ ( y, C U ) ≤ ρ ( x, C U ) − a for all y ∈ C U a \ C U .So, ρ ( C U a , C U ) ≤ ρ ( x, C U ) − a , and thus ξ ( x, U a ) = ρ ( x, C U a ) ≥ a .Finally, ξ ( y, U ) = ρ ( y, C U ) ≥ ρ ( x, C U ) − a = ξ ( x, U ) − a for all y ∈ U a because U a ⊂ W . Hence, ξ satisfies E (cid:3) Let consider the following condition, where ρ ( x, C ) is a non-negativefunction with C being a regularly closed subset of X :K1) ∗∗ For any regularly closed C $ X there is y C with ρ ( y, C ) > ρ ( x, C ) = 0 for all x ∈ C . Remark 2.3.
Observe that in the previous lemma we actually provedthe following more general statement: Suppose ρ satisfies conditions K ∗∗ and K − K ρ ( x, C ) ≤ x ∈ X and all regularlyclosed sets C ⊂ X . Then ξ ( x, U ) = sup { ρ ( x, C ) : C ∪ U = X } definesa π -capacity on X . Moreover, ξ is regular if ρ satisfies also K Corollary 2.4.
Suppose there is a function ρ on X satisfying condi-tions K ∗∗ and K − K . Then there is a quasi κ -metric d on X .Moreover, d is regular if ρ satisfies also condition K . uasi κ -metrics 5 Proof.
We can suppose that ρ is normed. Then, by Lemma 2.2, thereis a π -capacity ξ on X . Finally, Lemma 2.1, implies the existence of aquasi κ -metric d on X . Moreover, if ρ satisfies condition K ξ is regular, so is d . (cid:3) Theorem 2.5.
Any product of (regularly) quasi κ -metrizable spaces is(regularly) quasi κ -metrizable.Proof. Suppose X = Q α ∈ A X α and for every α there is a normed (reg-ular) quasi κ -metric ρ α on X . Following the proof of [7, Theorem 2],for every α let T α be the family of all regularly open subsets of X α and let B be the standard base for X consisting of sets of the form U = T ni =1 π − α i ( U i ) with U i ∈ T α i and U i = X α i , where π α : X → X α is the projection. Denote by v ( U ) the collection { α , ..., α n } . Accord-ing to Lemma 2.2, for every α there exists a (regular) π -capacity ξ α on X α . Consider the function ξ : X × B → R defined by ξ ( x, U ) =inf α ∈ v ( U ) ξ α ( π α ( x ) , π α ( U )) / | v ( U ) | . Obviously, condition E
1) is satisfied.Moreover, since for each α i the set W i = { z ∈ X α i : ξ i ( z, U i ) > } isopen and dense in U i , the set W = T ni =1 π − α i ( W i ) is open and dense in U . So, ξ satisfies condition E ξ is a (regular) capacity provide each ξ α is so, see the proof of [8,Theorem 15] and [7, Theorem 2]. The same arguments show that ξ also satisfies conditions E − E E
5) in case each ξ α is regular. Therefore, ξ is a (regular) π -capacity. Finally, by Lemma2.1, there exists a (regular) quasi κ -metric on X . (cid:3) Some more properties of quasi κ -metrizable spaces Proposition 3.1.
Let X be a quasi κ -metrizable space and Y ⊂ X .The Y is also quasi κ -metrizable in each of the following cases: (i) Y is dense in X ; (ii) Y is regularly closed in X ; (iii) Y is open in X .Proof. If ρ is a quasi κ -metric on X and Y ⊂ X is dense, the equality d ( y, U Y ) = ρ ( y, U X ), where U ⊂ Y is open, defines a quasi κ -metric on Y . The second case follows from the observation that every regularlyclosed subset of Y is also regularly closed in X . The third case is aconsequence of the first two because every open subset of X is densein its closure. (cid:3) Let consider the following condition.K4) ∗ ρ ( x, S C n ) = inf n ρ ( x, C n ) for every increasing sequence { C n } of regularly closed sets in X . Lemma 3.2.
Suppose X is a space admitting a non-negative function ρ ( x, C ) satisfying conditions K ∗ , K , K and K ∗ . Then X isquasi κ -metrizable provided X has countable cellularity. In particular,every compact space admitting such a function ρ is quasi κ -metrizable.Proof. It suffices to show that ρ satisfies condition K
4) in case X isof countable cellularity, and this follows from [3, Proposition 2.1]. Forreader’s convenience we provide a proof. Let { C α } α be an increasingtransfinite family of regularly closed sets in X . Then S α C α = S α U α and { U α } α is also increasing, where U α is the interior of C α . Since X has countable cellularity, there are countably many α i such that S i ≥ U α i is dense in S α U α . We can assume that the sequence { α i } isincreasing, so is the sequence { U α i } . Because ρ satisfies condition K ∗ ,we have ρ ( x, S C α i ) = inf i ρ ( x, C α i ). This implies that ρ ( x, S C α ) =inf α ρ ( x, C α ). Indeed, since S C α i = S C α , inf α ρ ( x, C α ) < inf i ρ ( x, C α i )for some x ∈ X would implies the existence of α with ρ ( x, C α ) < inf i ρ ( x, C α i ) for all i . Because any two elements of the family { C α } α are comparable with respect to inclusion, the last inequality meansthat C α contains all C α i . Hence, C α = S α C α and ρ ( x, C α ) would beequal to inf i ρ ( x, C α i ), a contradiction.It was shown in [9, Theorem 1.4] that every compact space X admit-ting a non-negative function ρ ( x, C ) satisfying conditions K ∗ , K K
3) and K ∗ is skeletally generated, and hence X has countable cel-lularity. Therefore, any such compactum is quasi κ -metrizable. (cid:3) It was shown by Chigogidze [1] that the ˇCech-Stone compactificationof every pseudocompact κ -metrizable space is κ -metrizable. We havea similar result for quasi κ -metrizable spaces. Theorem 3.3. If X is a pseudocompact (regularly) quasi κ -metrizablespace, then βX is (regularly) quasi κ -metrizable.Proof. Suppose ρ ( x, C ) is a quasi κ -metric on X . We can assume that ρ ( x, U X ) ≤ x ∈ X and all open U ⊂ X ( U X denotes theclosure of U in X ). For every open W ⊂ βX consider the function f W on X defined by f W ( x ) = ρ ( x, W ∩ X X ). Let e f W : βX → R bethe continuous extension of f W , and define d ( y, W ) = e f W ( y ), y ∈ βX .Obviously, d ( y, W ) = 0 if y ∈ W ∩ X . Since W ∩ X is dense in W , d ( y, W ) = 0 for all y ∈ W . Moreover, if W = βX , then W ∩ X X = X .So, there is an open dense subset V of X \ W with ρ ( x, W ∩ X X ) > x ∈ V . Since f W is continuous, the set e V = { y ∈ βX : f W ( y ) > } is open in βX and disjoint from W . Finally, because V ⊂ e V and V is uasi κ -metrics 7 dense in X \ W , e V is dense in βX \ W . So, d satisfies condition K ∗ .Conditions K
2) and K
3) also hold. Hence, by Lemma 3.2, it sufficesto show that d satisfies K ∗ . To this end, let { W n } be an increasingsequence of regularly closed subsets of βX and W = S n ≥ W n . Wehave d ( y, W ) ≤ inf n d ( y, W n ) for all y ∈ βX . Moreover, since ρ satisfies K d ( y, W ) = inf n d ( y, W n ) if y ∈ X . Suppose there is y ∈ βX \ X with d ( y , W ) < inf n d ( y , W n ). Consequently, for every n there existsa neighborhood V n of y in βX such that δ < d ( y, W n ) for all y ∈ V n ,where d ( y , W ) < δ < inf n d ( y , W n ). We also choose a neighborhood V of y with d ( y, W ) < δ for all y ∈ V . This implies that d ( y, W ) <δ ≤ inf n d ( y, W n ) provided y ∈ V = T n ≥ V ∩ V n . But V ∩ X = ∅ because X is pseudocompact. Thus, d ( y, W ) < inf n d ( y, W n ) for any y ∈ V ∩ X , a contradiction.It follows from the definition of d that it satisfies condition K ρ is regular. (cid:3) Corollary 3.4.
Every pseudocompact quasi κ -metrizable space X isskeletally generated.Proof. We already noted that every quasi κ -metrizable compactum isskeletally generated, see [9]. So, by Theorem 3.3, βX is skeletallygenerated. Finally, by [2] and [9], every dense subset of a skeletallygenerated space is also skeletally generated. (cid:3) Proposition 3.5.
Suppose f : X → Y is a perfect open surjection and X is (regularly) quasi κ -metrizable. Then Y is also (regularly) quasi κ -metrizable.Proof. Let ρ be a quasi κ -metric on X . Since f is open, f − ( U ) = f − ( U ) for any open U ⊂ Y . So, f − ( U ) is regularly closed set in X and we define d ( y, U ) = sup { ρ ( x, f − ( U )) : x ∈ f − ( y ) } . One can check that d satisfies conditions K
2) and K K
5) in case ρ is regular. Moreover, U = Y implies f − ( U ) = X . So,there is a dense open subset V ⊂ X \ f − ( U ) such that ρ ( x, f − ( U )) > x ∈ V . Then f ( V ) is a dense and open subset of Y \ U such that f − ( y ) ∩ V = ∅ for all y ∈ f ( V ). Hence, d ( y, U ) > y ∈ f ( V ). If y f ( V ), then f − ( y ) ∩ V = ∅ . Thus, d ( y, U ) > y ∈ f ( V ). Finally, letcheck continuity of the functions d ( · , U ). Suppose d ( y , U ) < ε for some y and U . Then ρ ( x, f − ( U )) < ε for all x ∈ f − ( y ). Consequently,there is a neighborhood W of f − ( y ) with ρ ( x, f − ( U )) < ε for all x ∈ W . Since, f is closed, y has a neighborhood G such that f − ( G ) ⊂ W . This implies that d ( y, U ) < ε for all y ∈ G . Now, let d ( y , U ) > δ forsome δ ∈ R . So, there exists x ∈ f − ( y ) with ρ ( x , f − ( U )) > δ .Choose a neighborhood O of x such that ρ ( x, f − ( U )) > δ for all x ∈ O . Then, f ( O ) is a neighborhood of y and d ( y, U ) > δ for any y ∈ f ( O ). Therefore, each d ( ., U ) is continuous. (cid:3) Recall that a surjective map f : X → Y is irreducible provided thereis no a proper closed subset F of X with f ( F ) = Y . Proposition 3.6.
Let f : X → Y be a perfect irreducible surjection,and Y is (regularly) quasi κ -metrizable. Then X is also (regularly)quasi κ -metrizable.Proof. Suppose ρ is a quasi κ -metric on Y . For every regularly closed C ⊂ X define d ( x, C ) = ρ ( f ( x ) , f ( C )). This definition is correct be-cause f ( C ) is regularly closed in Y . Indeed, let C = U with U openin X . Since f perfect and irreducible, we have f ( C ) = U ♯ , where U ♯ = { y ∈ Y : f − ( y ) ⊂ U } = Y \ f ( X \ U ) is open in Y . It is easilyseen that d satisfies conditions K
2) and K K
4) followsfrom the equality f ( S α C α ) = S α f ( C α ) for any family of regularlyclosed sets in X . To see that d satisfies also condition K ∗ , we ob-serve that for every regularly closed C $ X there is a dense open subset V ⊂ Y \ f ( C ) such that ρ ( y, f ( C )) > y ∈ V . Then W = f − ( V )is open in X and disjoint from C . Moreover, d ( x, C ) > x ∈ W .It remains to show that W is dense in X \ C . And that is really truebecause for every open O ⊂ X \ C the set O ♯ is a non-empty opensubset of Y \ f ( C ). So, O ♯ ∩ V = ∅ , which implies W ∩ O = ∅ .One can also see that d is regular provided so is ρ . (cid:3) It is well known that for every space X there is a unique extremallydisconnected space e X and a perfect irreducible map f : e X → X . Thespace e X is said to be the absolute of X . A space Y is called co-absoluteto X if their absolutes are homeomorphic. Corollary 3.7.
The absolute of any (regularly) quasi κ -metrizable spaceis (regularly) quasi κ -metrizable. Remark 3.8.
The last corollary shows that the class of κ -metrizablespaces is a proper subclass of the quasi κ -metrizable spaces. Indeed,let X be a κ -metrizable compact infinite space. Then its absolute aX is quasi κ -metrizable. On the other hand, aX being extremallydisconnected can not be κ -metrizable (otherwise, it should be discreteby [8, Theorem 11]). Corollary 3.9.
Every compact space co-absolute to a quasi κ -metrizablespace is skeletally generated. uasi κ -metrics 9 Proof.
Let X and Y be compact spaces having the same absolute Z .So, there are perfect irreducible surjections g : Z → Y and f : Z → X .If Y is quasi κ -metrizable, then so is Z , see Proposition 3.6. Hence, Z is skeletally generated, and by [5, Lemma 1], X is also skeletallygenerated. (cid:3) Recall that the hyperspace expX consists of all compact non-emptysubsets F of X such that the sets of the form[ U , .., U k ] = { H ∈ expX : H ⊂ k [ i=1 U i and H ∩ U i = ∅ for all i } form a base B exp for expX, where each U i belongs to a base B for X ,see [6]. Proposition 3.10. If X is (regularly) quasi κ -metrizable, so is expX .Proof. Let B be a base for X and ρ be a (regular) quasi κ -metricon X . Then ρ generates a (regular) π -capacity ξ ρ : X × B → R on X . Following the proof of [7, Theorem 3], we define a function ξ : expX × B exp → R by ξ ( F, [ U , .., U k ]) = 1 k min { inf x ∈ F max i ξ ρ ( x, U i ) , min i sup x ∈ F ξ ρ ( x, U i ) } . It was shown in [7] that ξ satisfies conditions E E
3) and E ξ is regular provided ξ ρ is regular. Let show that ξ satis-fies condition E ξ is satisfies E U , .., U k ] there is a dense subset V exp ⊂ [ U , .., U k ] with ξ ( F, [ U , .., U k ]) > F ∈ V exp . To this end, for each i fix anopen dense subset V i of U i such that ξ ρ ( x, U i ) > x ∈ V i . Let V exp consists of all finite sets F ⊂ X such that F ⊂ S ni =1 V i and F ∩ V i = ∅ for all i . Then V exp is dense in [ U , .., U k ] and ξ ( F, [ U , .., U k ]) > F ∈ V exp . Hence, by Lemma 2.1, expX is (regularly) quasi κ -metrizable. (cid:3) Shchepin [7, Theorem 3a] has shown that if expX is κ -metrizable,then so is X . We don’t know if a similar result is true for quasi κ -metrizable spaces.4. Skeletally generated spaces
In this section we provide a characterization of skeletally generatedcompact spaces in terms of functions similar to quasi κ -metrics. Wesay that a non-negative function d : X × C → R is a weak κ -metric ,where C is the family of all regularly closed subsets of X , if it satisfiesconditions K ∗ , K − K
3) and the following one: K For every increasing transfinite family { C α } α ⊂ C the function f ( x ) = inf α d ( x, C α ) is continuous. Theorem 4.1.
A compact space is skeletally generated if and only ifit is weakly κ -metrizable.Proof. First, let show that every skeletally generated compactum X isweakly κ -metrizable. We embed X as a subset of R τ for some cardinal τ . Then, according to [9, Theorem 1.1], there is a function e : T X → T R τ between the topologies of X and R τ such that: ( i ) e(U) ∩ e(V) = ∅ provided U and V are disjoint; ( ii ) e(U) ∩ X is dense in U . We definea new function e : T X → T R τ ,e ( U ) = [ { e( V ) : V ∈ T X and V ⊂ U } . Obviously e satisfies conditions ( i ) and ( ii ), and it is also monotone,i.e. U ⊂ V implies e ( U ) ⊂ e ( V ). Moreover, for every increasingtransfinite family γ = { U α } of open sets in X we have e ( S α U α ) = S α e ( U α ). Indeed, if z ∈ e ( S α U α ), then there is an open set V ∈ T X with V ⊂ S α U α and z ∈ e( V ). Since V is compact and the family isincreasing, V is contained in some U α . Hence, z ∈ e( V ) ⊂ e ( U α ).Consequently, e ( S α U α ) ⊂ S α e ( U α ). The other inclusion followsfrom monotonicity of e .Because R τ is κ -metrizable (see [7]), there is a κ -metric ρ on R τ . Forevery regularly closed C ⊂ X and x ∈ X we can define the function d ( x, C ) = ρ ( x, e (IntC)), where e ( U ) is the closure of e ( U ) in R τ . Itis easily seen that d satisfies conditions K − K K and K ∗ ). Indeed, assume { C α } is an increasingtransfinite family of regularly closed sets in X . We put U α = Int C α forevery α and U = S α U α . Thus, e ( U ) = S α e ( U α ). Since { e ( U α ) } isan increasing transfinite family of regularly closed sets in R τ , for every x ∈ X we have ρ ( x, [ α e ( U α )) = inf α ρ ( x, e ( U α )) = inf α d ( x, C α ) . Hence, the function f ( x ) = inf α d ( x, C α ) is continuous on X because sois ρ ( ., S α e ( U α )). To show that K ∗ ) also holds, observe that d ( x, C ) =0 if and only if x ∈ X ∩ e (IntC). Because e (IntC) ∩ X is dense in C , C ⊂ e (IntC). Hence, V = X \ e (IntC) is contained in X \ C and d ( x, C ) > x ∈ V . To prove V is dense in X \ C , let x ∈ X \ C and W x ⊂ X \ C be an open neighborhood of x . Then W ∩ IntC = ∅ , soe ( W ) ∩ e (IntC) = ∅ . This yields e ( W ) ∩ X ⊂ V . On the other hand,e ( W ) ∩ X is a non-empty subset of W , hence W ∩ V = ∅ . Therefore, d is a weak κ -metric on X . uasi κ -metrics 11 The other implication was actually established in the proof of The-orem 1.4] from [9], and we sketch the proof here. Suppose d is a weak κ -metric on X and embed X in a Tychonoff cube I A with uncountable A , where I = [0 , B ⊂ A let A B be a count-able base for X B = π B ( X ) consisting of all open sets in X B of the form X B ∩ Q α ∈ B V α , where each V α is an open subinterval of I with rationalend-points and V α = I for finitely many α . Here π B : I A → I B denotesthe projection, and let p B = π B | X . For any open U ⊂ X denote by f U the function d ( · , U ). We also write p B ≺ g , where g is a map definedon X , if there is a map h : p B ( X ) → g ( X ) such that g = h ◦ p B . Since X is compact this is equivalent to the following: if p B ( x ) = p B ( x )for some x , x ∈ X , then g ( x ) = g ( x ). We say that a countable set B ⊂ A is d -admissible if p B ≺ f p − B ( V ) for every V ∈ A <ωB . Here A <ωB is the family of all finite unions of elements from A B . Denote by D the family of all d -admissible subsets of A . We are going to show thatall maps p B : X → X B , B ∈ D , are skeletal and the inverse system S = { X B : p BD : D ⊂ B, D, B ∈ D} is σ -continuous. Since X = lim ← S ,this would imply that X is skeletally generated, see [9] and [10].Following the proof of [9, Theorem 1.4], one can show that for anycountable set B ⊂ A there is D ∈ D with B ⊂ D , and the union ofany increasing sequence of d -admissible sets is also d -admissible. So, weneed to show only that p B : X → X B is a skeletal map for every B ∈ D .Suppose there is an open set U ⊂ X such that the interior in X B of p B ( U ) is empty. Then W = X B \ p B ( U ) is dense in X B . Let { W m } m ≥ be a countable cover of W with W m ∈ A B for all m . Since A <ωB isfinitely additive, we may assume that W m ⊂ W m +1 , m ≥
1. Because B is d -admissible, p B ≺ f p − B ( W m ) for all m . Hence, there are contin-uous functions h m : X B → R with f p − B ( W m ) = h m ◦ p B , m ≥
1. Re-call that f p − B ( W m ) ( x ) = d ( x, p − B ( W m )) and p − B ( W ) = S m ≥ p − B ( W m ).Therefore, f p − B ( W ) ( x ) = d ( x, p − B ( W )) ≤ f ( x ) = inf m f p − B ( W m ) ( x ) for all x ∈ X . Moreover, f is continuous and f p − B ( W m +1 ) ( x ) ≤ f p − B ( W m ) ( x ) be-cause W m ⊂ W m +1 . The last inequalities together with p B ≺ f p − B ( W m ) yields that p B ≺ f . So, there exists a continuous function h on X B with f ( x ) = h ( p B ( x )) for all x ∈ X . But f ( x ) = 0 for all x ∈ p − B ( W ), so f ( p − B ( W )) = 0. This implies that h ( W ) = 0. Since p B ( p − B ( W )) = W = X B , we have that h is the constant functionzero. Consequently, f ( x ) = 0 for all x ∈ X . Finally, the inequality d ( x, p − B ( W )) ≤ f ( x ) yields that d ( x, p − B ( W )) = 0 for all x ∈ X . Onthe other hand, p − B ( W ) ∩ U = ∅ . So, according to K ∗ ), there is an open subset U ′ of U with d ( x, p − B ( W )) > x ∈ U ′ , acontradiction. (cid:3) Because any compactification of a skeletally generated space is skele-tally generated (see [9]) and the weakly κ -metrizability is a hereditaryproperty with respect to dense subsets, we have the following Corollary 4.2.
Every skeletally generated space is weakly κ -metrizable. All results in Section 3, except Proposition 3.10, remain valid forweakly κ -metrizable spaces. Theorem 4.1 and a result of Kucharski-Plewik [5, Theorem 6] imply that Proposition 3.10 is also true forweakly κ -metrizable compacta. But the following questions are stillopen. Question 4.3.
Is any product of weakly κ -metrizable spaces weakly κ -metrizable?If there exists a counter example to Question 4.3 which, in addi-tion has a countable cellularity, then next question would have also anegative answer. Question 4.4.
Is any weakly κ -metrizable space with a countable cel-lularity skeletally generated? Acknowledgments.
The author would like to express his gratitudeto A. Kucharski for his careful reading and suggesting some improve-ments of the paper.
References [1] A. Chigogidze, On κ -metrizable spaces , Uspehi Mat. Nauk (1982), no. 2,241–242 (in Russian).[2] P. Daniels, K. Kunen and H. Zhou, On the open-open game , Fund. Math. (1994), no. 3, 205–220.[3] A. Kucharski and S. Turek,
A generalization of κ -metrizable spaces ,arXiv:161208838.[4] A. Kucharski and S. Plewik, Skeletal maps and I -favorable spaces , Acta Univ.Carolin. Math. Phys. (2010), 67–72.[5] A. Kucharski and S. Plewik, Game approach to universally Kuratowski-Ulamspaces , Topology Appl. (2007), no. 2, 421–427.[6] K. Kuratowski,
Topology, vol. I , Academic Press, New York; PWN-PolishScientific Publishers, Warsaw 1966.[7] E. Shchepin, On κ -metrizable spaces , Math. USSR Izvestija (1980), no. 2,1–34.[8] E. Shchepin, Topology of limit spaces of uncountable inverse spectra , RussianMath. Surveys (1976), 155–191.[9] V. Valov,
I-favorable spaces: Revisited
Topology Proc. (2018), 277–292. uasi κ -metrics 13 [10] V. Valov, External characterization of I -favorable spaces , MathematicaBalkanica (2011), no. 1-2, 61–78 Department of Computer Science and Mathematics, Nipissing Uni-versity, 100 College Drive, P.O. Box 5002, North Bay, ON, P1B 8L7,Canada
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