Mahavier products indexed by partial orders
aa r X i v : . [ m a t h . GN ] J a n Mahavier products indexed by partial orders
Steven Clontz and Jacob DunhamJanuary 7, 2021
Abstract
Let X be separable metrizable, and let f ⊆ X be a non-trivial relation on X . For agiven partial order h P, ≤i , the Mahavier product M h X, f, P i ⊆ X P collects functionssuch that x ( p ) ∈ f ( x ( q )) for all p ≤ q . Clontz and Varagona previously showed forwell orders P that M h X, f, P i is separable metrizable exactly when P is countable; weextend this result to hold for all partial orders. Let X be a topological space, let f ⊆ X be a relation on X , and let Q be a set preorderedby (reflexive and transitive) (cid:22) .Extending work done in e.g. [1, 2] we consider the subspace M h X, f, Q i of X Q where x ( p ) ∈ f ( x ( q )) for all p (cid:22) q . Such subspaces are often known as generalized inverse limits,or as we will refer to them, Mahavier products. For an introduction to such structures,often considered in the context of continuum theory, we direct the reader to [3]. Much ofthe literature on this subject considers only simple indices, particularly P = N or Z with itsusual order.It’s immediate that any subspace of X Q is separable metrizable whenever Q is count-able. In [4, 5] the authors consider whether M h X, f, Q i might be metrizable when Q is anuncountable well-order. It turns out that, except in trivial situations, the answer is no.We aim to extend this result to the more general case where P is any partial order, thatis, any antisymmetric preorder. To do this, we reintroduce the notion of a partially-orderedtopological space (POTS) originally defined in [6], a generalization of the class of linearlyordered topological spaces (LOTS) studied broadly throughout general topology. For convenience we formally define the notion of a preorder and partial order here.
Definition 1. A preorder h Q, (cid:22)i is a set paired with a reflexive ( x (cid:22) x ) and transitive( x (cid:22) y (cid:22) z ⇒ x (cid:22) z ) relation.We adopt the convention of using Q for preorders as they are sometimes known as quasi-orders . Definition 2. A partial order h P, ≤i is a preorder that is antisymmetric ( x < y ⇒ y < x ).We note that every preorder admits a natural partial order.1 roposition 3. Let Q be preordered by (cid:22) , and set p ∼ q if and only if p (cid:22) q and q (cid:22) p .Then ∼ is an equivalence relation, and its set of equivalence classes P = { [ p ] : p ∈ Q } ordered by A ≤ B if and only if p (cid:22) q for all p ∈ A, q ∈ B is a partial order. Notation 4.
For each preorder h Q, (cid:22)i we denote the partial order given in the previousproposition by h Q ⋆ , ≤i .As we will see, for our purposes we can use the partial order Q ⋆ in place of any preorder Q , so we now only consider partial orders. Notation 5.
Let h P, ≤i be a partial order. Then we adopt the following notation similarto that commonly used for linear orders; for example: ( ← , p ) = p < = { r ∈ P : r < p } [ p, → ) = p ≥ = { r ∈ P : r ≥ p } ( p, q ] = p > ∩ q ≤ p = P \ p ≤ Of course, if a partial order is total , that is, p ≤ q or q ≤ p for all p, q , then we havethe usual idea of a total order or linear order . A wide class of topological spaces known as LOTS are defined in terms of linear orders; we characterize them for all partial orders asfollows.
Definition 6. A partially ordered topological space , or POTS , is a topological space X partially ordered by ≤ with a subbasis { x : x ∈ X } ∪ { x : x ∈ X } .The reader may verify that if ≤ is a linear order, then this subbasis yields the usual basisof open intervals { ( x, y ) : x, y ∈ X ∪ {← , →}} . However, the next example illustrates thatthis does not hold for general POTS; indeed, a POTS may not even be Hausdorff (thoughit will always be T ). Example 7.
Let [ ω ] < ℵ = { F ⊆ ω : F is finite } be partially ordered by ⊆ .We then let F ⊆ ω be finite, and thus for any x F , the singleton set { x } ∈ F .Consider now the subbasic open set F . If F = ∅ then certainly F = ∅ as well, and if F = { y } then { x } ∈ F for all x ∈ ω \ { y } . Finally, if | F | > , then { x } ∈ F for all x ∈ ω .Thus, any two subbasic open sets have infinite intersection, showing (cid:10) [ ω ] < ℵ , ⊆ (cid:11) cannot beHausdorff.Ward [6] gives the following definition for continuous partial orders; some authors [7]require this as a condition of being a “pospace”. Definition 8.
A POTS is said to be continuous if for each p q there exist open sets U, V such that p ∈ U, q ∈ V and r s for all r ∈ U, s ∈ V .Note that the definition immediately implies the Hausdorff property, and may be rechar-acterized as follows. Definition 9.
A subset A of a partial order is said to be downward if for all p ∈ A , p ≤ ⊆ A .A subset A of a partial order is said to be upward if for all p ∈ A , p ≥ ⊆ A . Lemma 10 ([6]) . The following are equivalent for a given POTS h P, ≤i . P is continuous. • {h x, y i : x ≤ y } ⊆ P is closed. • For each p q , there exist disjoint open sets U, V such that p ∈ U, q ∈ V , U is upward,and V is downward. We may define the Mahavier product as a subspace of the usual Tychonoff product. Byconvention, we will treat relations f ⊆ X as set-valued functions, that is, f ( x ) = { y : h x, y i ∈ f } . Definition 11.
Let X be a topological space. A relation f ⊆ X is said to be a V -relation if it is closed, idempotent ( f ( x ) = f ( x ) ), surjective ( ∀ y ∈ X ∃ x ∈ X ( y ∈ f ( x )) ), and serial( ∀ x ∈ X ∃ y ∈ X ( y ∈ f ( x )) ). Definition 12.
Let X be a topological space, f ⊆ X be a V -relation, and Q be a preorder.Then M h X, f, Q i = { x ∈ X Q : x ( p ) ∈ f ( x ( q )) for all p (cid:22) q } is the Mahavier product , considered as a subspace of the usual Tychonoff product X Q .This definition may be contrasted with other interesting subspaces of X Q studied ingeneral topology, such as the Σ -product. (We pose a question relating to compact subspacesof the Σ -product, known as Corson compacta, at the end of the paper.)In [5], the first author and Varagona showed the following. Definition 13.
A relation f ⊆ X satisfies condition Γ if there exist distinct x, y ∈ X suchthat h x, x i , h x, y i , h y, y i ∈ f . Lemma 14.
Let X be weakly countably compact and f = ι = {h x, x i : x ∈ X } be a V -relation. Then f satisfies condition Γ . To this end, we will concentrate on the following particular Mahavier product.
Definition 15.
Let X contain distinct elements , . Then γ X ⊆ X satisifes γ X (0) = X and γ X ( x ) = { } otherwise.Of course, γ X is a V -relation that satisfies condition Γ . If X = 2 = { , } , we simplywrite γ . Proposition 16.
Let Y ⊆ X and g ⊆ f . Then M h Y, g, Q i ⊆ M h X, f, Q i .In particular, let ⊆ X and γ ⊆ f . Then M h , γ, Q i ⊆ M h X, f, Q i . Recalling our motivation, we want to characterize the separable metrizability of M h X, f, P i (for separable metrizable X and f satisfying condition Γ ). Since M h X, f, P i is trivially sep-arable and metrizable when P is countable, we will consider the case for uncountable P . Inparticular, we will show that the subspace M h , γ, P i fails to be second-countable in thiscase.First consider the following simplification for preorders that allows us to concentrate onpartial orders. We use the symbol ∼ = to denote homeomorphic spaces.3 roposition 17. Let Q be a preorder. Then M h , γ, Q i ∼ = M h , γ, Q ⋆ i .Proof. Note that for any x ∈ M h , γ, Q i , if x ( p ) = n then x ( q ) = n for all q ∼ p , as q ≤ p ≤ q . Thus, the function f defined by f ( x )([ p ]) = x ( p ) is well defined, and the readermay verify it is in fact a homeomorphism between M h , γ, Q ∗ i and M h , γ, Q i .Several properties of M h , γ, P i are inherited from its superspace P ; it’s regular andHausdorff for instance. The reader may also confirm that it is a closed subspace of P ,showing compactness.To most elegantly demonstrate that M h , γ, P i is second-countable exactly when P iscountable, we will investigate a new POTS related to P that it is homeomorphic to. Recall the well-known compactification ˆ L = { A ⊆ L : A is closed and downward } of a givenLOTS L . The requirement that A be closed is necessary for { l ≤ : l ∈ L } ∼ = L to be dense in ˆ L . But by dropping that requirement, we will obtain our desired copy of M h , γ, P i . Definition 18.
Let P be a partial order. Then we denote the POTS of downward subsets by ˇ P = { A ⊆ P : A is downward } . Proposition 19.
Let P be a partial order. Then (cid:10) ˇ P , ⊆ (cid:11) is a continuous POTS.Proof. It’s immediate that ⊆ is a partial order. To see that it is continuous on ˇ P , we willverify bullet 3 of Lemma 10. Let X, Y ∈ ˇ P with X Y and x ∈ X \ Y , Note x ≤ , x ∈ ˇ P ,and furthermore X ∈ ( x ) and Y ∈ ( x ≤ ) . We will now show ( x ) is upward and ( x ≤ ) is downward. Let X ∈ ( x ) with X ⊇ X . As X ∈ ( x ) , X x . Thus, X x and so X ∈ ( x ) . Now let Y ∈ ( x ≤ ) with Y ⊆ Y . As Y ∈ ( x ≤ ) , Y x ≤ .Therefore, Y x ≤ , showing that Y ∈ ( x ≤ ) . We finally show these two open sets aredisjoint. ( x ≤ ) ∩ ( x ) is the collection of downward sets which are not supersets of x ≤ and not subsets of P \ x ≥ . Any such set must somehow miss some y ≤ x , yet contain some z ≥ x , contradicting the property that sets in ˇ P are downward. Thus, ( x ≤ ) ∩ ( x ) mustbe empty. Remark . There is an obvious bijection between elements of M h , γ, P i and ˇ P . Eachsequence x ∈ M h , γ, P i satsifies the property that x ( q ) = 1 implies x ( p ) = 1 for all p ≤ q ,showing that { p ∈ P : x ( p ) = 1 } is downward. Likewise, the characteristic function for eachdownward set A ∈ ˇ P belongs to M h , γ, P i .To see that this bijection is actually a homeomorphism, we first introduce a convenientbasis for ˇ P as an alternative to its POTS subbasis. Proposition 21.
Let B h T, F i = { A ∈ ˇ P : T ⊆ A and F ∩ A = ∅} . Then { B h T, F i : T, F ⊆ finite P } is a basis for a topology on ˇ P . Theorem 22.
The basis given in the previous proposition induces the POTS topology.Proof.
Consider
T, F ⊆ finite P . Let A ∈ B h T, F i , so for any t ∈ T and f ∈ F , A ∈ ( t ) ∩ ( f ≤ ) . Therefore, A ∈ T ( { ( t ) : t ∈ T } ∪ { ( f ≤ ) : f ∈ F } ) . Now let A ∈ T ( { ( t ) : t ∈ T } ∪ { ( f ≤ ) : f ∈ F } ) . Then certainly t ∈ A for all t ∈ T , and similarly f A for all f ∈ F . Thus, A ∈ B h T, F i , showing that \ ( { ( t ) : t ∈ T } ∪ { ( f ≤ ) : f ∈ F } ) = B h T, F i . A ∈ R for some R ∈ ˇ P , so A ∈ ˇ P \ R ⊆ . Therefore, there exists some r ∈ A with r R , and so A ∈ B h{ r } , ∅i ⊆ R . Similarly, if A ∈ R , then there exists some r ∈ R with R A , showing that A ∈ B h∅ , { r }i ⊆ R .Therefore, the POTS subbasis and the basis of B h T, F i s both generate the same topol-ogy. Corollary 23. ˇ P ∼ = M h , γ, P i .Proof. The map h : ˇ P → M h , γ, P i defined by h ( A )( p ) = ( p A p ∈ A was shown to be a bijection in Remark 20. We now consider h [ B h T, F i ] . As T ⊆ A and F ∩ A = ∅ for all A ∈ B h T, F i , h will map the set B h T, F i to the set of sequences which arerestricted to { } on T , and { } on F . Thus, let U p = { } if p ∈ T , U p = { } if p ∈ F , and U p = { , } otherwise. Then h [ B h T, F i ] = Q p ∈ P U p , showing h is a homeomorphism.So ˇ P is a compact Hausdorff subspace of M h X, f, P i , provided f satisfies condition Γ .Thus the following theorem completes our characterization. Theorem 24. ˇ P is second-countable if and only if P is countable.Proof. If P is countable, P is second-countable, and therefore its subspace M h , γ, P i ∼ = ˇ P is second-countable.If P is uncountable, let B be a basis for ˇ P . Suppose P is uncountable and B is a basisfor P . Then for all p ∈ P , fix B p ∈ B such that p ≤ ∈ B p ⊆ B h{ p } , ∅i . Now let p = q andwithout loss of generality assume p q . Therefore, q ≤ B h{ p } , ∅i , and so q ≤ B p . Byconstruction we have q ≤ ∈ B q , and so it must be that B p = B q . Thus, each B p is unique,and so B must be uncountable, showing that ˇ P is not second-countable. Corollary 25.
Let X be separable metrizable, f satisfy condition Γ , and P be a partialorder. Then M h X, f, P i is separable metrizable if and only if P is countable.Proof. If P is countable, X P is separable metrizable, and therefore its subspace M h X, f, P i is separable metrizable.If P is uncountable, M h X, f, P i contains a non-second-countable subspace ˇ P , and there-fore cannot be second-countable itself.Note that this corollary is a generalization of Theorem 5.1 of [5], as ˇ α = α + 1 for allordinals α .Of course this corollary cannot extend to preorders h Q, (cid:22)i : if p (cid:22) q and q (cid:22) p for all p, q ∈ Q , then M h , γ, Q i ∼ = 2 regardless of the cardinality of Q . But in this case, Q ⋆ = { Q } is countable, so the second-countability of M h , γ, Q i ∼ = M h , γ, Q ⋆ i was guaranteed.However, this does not say anything about M h X, f, Q i , since Q is not second-countablewhen Q is uncountable. Question 26.
Let X be separable metrizable, f satisfy condition Γ , and Q be a preorder.Is M h X, f, Q i separable metrizable if and only if Q ⋆ is countable?
5n [5] it was also observed that the Corson compactness of M h X, f, α i was linked tothe countability of the ordinal α (as the same holds for ˇ α = α + 1 ). But the followinggeneralization remains open. Question 27.
Let P be a partial (or linear) order. Is ˇ P Corson compact if and only if P is countable? If so, then for Corson compact X and f satisfying condition Γ , M h X, f, P i would beCorson compact if and only if P is countable. References [1] Sina Greenwood and Judy Kennedy. Connected generalized inverse limits over intervals.
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