Uniformly joinable, locally uniformly joinable, and weakly chained uniform spaces
aa r X i v : . [ m a t h . G M ] J a n UNIFORMLY JOINABLE, LOCALLY UNIFORMLY JOINABLE,AND WEAKLY CHAINED UNIFORM SPACES
BRENDON LABUZ
Abstract.
Brodskiy, Dydak, LaBuz, and Mitra introduced the concepts ofuniform joinability and local uniform joinability for uniform spaces when de-veloping their theory of generalized uniform covering maps which was moti-vated by a paper by Berestovskii and Plaut. (Local) uniform joinability can bethought of as analogous to (local) path connectedness. A chain connected lo-cally uniformly joinable uniform space is uniformly joinable. This note gives anexample of a metric space that is uniformly joinable but not locally uniformlyjoinable.Plaut recently defined the concept of a weakly chained uniform space. Weshow that a weakly chained metrizable uniform space is locally uniformly join-able. Since local uniform joinability is equivalent to pointed 1-movability formetric continua, we find that weak chainability is equivalent to pointed 1-movability for such spaces.
Contents
1. Introduction 12. The Texas circle 23. Weakly chained spaces 3References 41.
Introduction
The concepts of uniform joinability and local uniform joinability were definedin [1] in the context of creating generalized uniform covering maps. Traditionalcovering space theory does not apply to locally complicated spaces. For such spacesone can consider chains instead of paths, and the inverse limit of finer and finerchains gives generalized paths. These chains require a way of measuring closenessacross a space so the natural venue for such a construction is the category of uniformspaces.Given a uniform space X and an entourage E of X , an E -chain in X is a sequenceof points x , . . . , x n in X such that ( x i , x i +1 ) ∈ E for each i < n . We define E -homotopies of E -chains by appealing to Rips complexes. Given an entourage E of X the Rips complex R( X, E ) is the subcomplex of the full complex over X whosesimplices are finite E -bounded subsets of X . A subset A of X is called E -boundedif for each x, y ∈ A , ( x, y ) ∈ E . Equivalently, A × A ⊂ E . Any E -chain x , . . . , x n determines a path in R ( X, E ) by simply joining successive terms x i , x i +1 by anedge path, a path along the edge joining x i and x i +1 . Then E -homotopies between Date : January 20, 2021. b b b
Figure 1.
The Texas circle E -chains can be defined in terms of homotopies between paths in R( X, E ). Two E -chains starting at the same point and ending at the same point are said to be E -homotopic if the corresponding paths in R( X, E ) are homotopic relative endpoints.This homotopy in R(
X, E ) can be chosen to be simplicial. Thus, two E -chains are E -homotopic if one can move from one to the other by a finite sequence of theaddition or deletion of a point in the E -chain (while keeping the endpoints fixed).Given an E -chain c , denote the E -homotopy class of c as [ c ] E .A generalized path is a collection of homotopy classes of chains α = { [ c E ] E } E where E runs over all entourages of X and for any F ⊂ E , c F is E -homotopic to c E .We write α E = [ c E ] E . Inverses and concatenations of generalized paths are definedin the obvious way. The set of generalized paths in X is denoted as GP( X ). Wealso have the pointed version GP( X, x ) of all generalized paths in X starting atsome basepoint x . Given an entourage E of X , define E ∗ to be the set of all pairs( α, β ) of elements of GP ( X, x ) such that α − E ∗ β E is E -homotopic to the E -chain x, y where x is the endpoint of α and y is the endpoint of β . Such a generalizedpath is called E -short. The set of all E ∗ as E runs over all entourages of X formsa basis for a uniform structure on GP( X, x ).A uniform space X is called uniformly joinable if every pair of points in X canbe joined by a generalized path. Equivalently, the endpoint map GP( X, x ) → X issurjective. Note any path in X induces a generalized path so if X is path connectedthen it is uniformly joinable. A uniform space X is called locally uniformly joinableif for each entourage E of X there is an entourage F of X such that if ( x, y ) ∈ F then x and y can be joined by an E -short generalized path.A metric space X induces a uniform structure as follows. Given ǫ > E ǫ isdefined to be the set of pairs ( x, y ) of points in X such that d( x, y ) ≤ ǫ . Then theset of all E ǫ form a basis for the uniform structure on X .For metric continua, by a theorem of Krasinkiewicz and Minc, uniform join-ability and local uniform joinability are equivalent (see [1, Corollary 6.6]). Anychain connected locally uniformly joinable uniform space is uniformly joinaable [1,Proposition 4.3] and it is easy to see that there are spaces that are locally uniformlyjoinable but not uniformly joinable (consider parallel lines in the plane). We givean example of a metric space that is uniformly joinable but not locally uniformlyjoinable. It is a path connected subset of the plane with the uniform structureinduced by the usual metric. 2. The Texas circle
Let T be the union of the graph of f ( x ) = sin x + 1 /x over [ π, ∞ ), the x -axisfrom π to ∞ , and the vertical line segment from ( π,
0) to ( π, /π ) (see Figure2). We call T the Texas circle as it can be considered a large scale version of the Since the identity function K w → K m , K a simplicial complex, from K equipped with theCW (weak) topology to K equipped with the metric topology is a homotopy equivalence, it doesnot matter which topology is chosen for R( X, E ). NIFORMLY JOINABLE, LOCALLY UNIFORMLY JOINABLE, AND WEAKLY CHAINED 3
Warsaw circle. Give T the uniform structure induced by the usual metric on theplane. Then T is path connected so it is uniformly joinable. Let us see that T isnot locally uniformly joinable. Suppose to the contrary that T is locally uniformlyjoinable. Then there is an entourage F so that if ( x, y ) ∈ F , x and y can be joinedby an E / -short generalized path. Let n ∈ N be such that if d( x, y ) ≤ /nπ then( x, y ) ∈ F . Consider the points x = ( nπ, /nπ ) and y = ( nπ, x, y ) ∈ F so there is an E / -short generalized path joining them. But the only generalizedpath joining them is the one induced by the path that travels along the graph of f all the way to the vertical line segment and then back along the x -axis and thisgeneralized path is not E / -short (notice a square of side length 1 / x and y , consider thatfor m > n and ǫ ≤ /mπ , the only E ǫ -chains joining x and y are ones that travelalong the graph of f all the way to the vertical line segment and then back along the x -axis, and the ones that travel to the right of x to at least the point ( mπ, /mπ )and then hop down to the x -axis and back to y (or some combination of these twotypes of E ǫ -chains). Now for these types of E ǫ -chains to form a generalized path weneed that for m ′ > m , the E /m ′ π -chain is E /mπ -homotopic to the E /mπ -chain.But that cannot always be the case. For fix an E /mπ -chain c . This chain has aright-most point, say with x -value x m . Choose m ′ so that ( m ′ − π > x m . Thenany E /m ′ π -chain joining x to y either travels through the vertical line segment ormust travel to at least ( m ′ π, /m ′ π ) before hopping down to the x -axis and thesetypes of E /m ′ π -chains are not E /mπ -homotopic to c .3. Weakly chained spaces
A uniform space is called chain connected if for every entourage E , any pair ofpoints can be joined by an E -chain. In the preprint of Plaut [2], a uniform spaceis defined to be weakly chained if it is chain connected and for any entourage E of X there exists an entourage F ⊂ E such that for any ( x, y ) ∈ F , there existsarbitrarily fine chains c joining x and y such that [ c ] E = [ x, y ] E (we can say thatthe chain is E -short). Clearly a chain connected locally uniformly joinable space isweakly chained. We will see that the converse holds for metrizable spaces. Proposition 3.1.
Let X be a metrizable uniform space. If X is weakly chainedthen it is locally uniformly joinable.Proof. Let a metric be given that induces the uniform structure on X . For n > E /n is the entourage consisting of pairs of points of X that are distance atmost 1 /n apart. Let E be an entourage of X . We will find an entourage F of X so that if ( x, y ) ∈ F , x and y can be joined by an E -short generalized path.Let F ⊂ E be an entourage such that if ( x, y ) ∈ F , x and y can be joined byarbitrarily fine chains that are E -short. Let F ⊂ E / be an entourage such thatif ( x, y ) ∈ F , x and y can be joined by arbitrarily fine chains that are F -short.Inductively choose, for each n ≥
3, an entourage F n ⊂ E /n such that if ( x, y ) ∈ F n , x and y can be joined by arbitrarily fine chains that are F n − -short.Suppose ( x, y ) ∈ F . We will inductively define F n -chains from x to y that willcreate a generalized path that is E -short. Let c be an F -chain joining x and y that is F -short. We define [ c ] F to be the F -level of the generalized path we areconstructing. Note [ c ] F = [ x, y ] F so c is E -short. BRENDON LABUZ
Now join pairs of successive points of c by F -chains that are F -short. Let c be the F -chain resulting from concatenating those chains. We define [ c ] F tobe the F -level of the generalized path we are constructing. Note [ c ] F = [ c ] F .Using induction, for each n ≥
3, join pairs of successive points of c n − by F n +2 -chains that are F n -short. Let c n be the F n +2 -chain resulting from concatenatingthose chains. We define [ c n ] F n to be the F n -level of the generalized path we areconstructing. Note [ c n ] F n = [ c n − ] F n .These equivalence classes induce a generalized path from x to y that is E -short. (cid:3) Recall that a chain connected locally uniformly joinable uniform space is uni-formly joinable ([1, Proposition 4.3]). Thus a weakly chained uniform space is uni-formly joinable. Also, according to [1, Corollary 6.6], for metric continua, the con-ditions of uniformly joinable, locally uniformly joinable, and pointed 1-movable areequivalent. Since metric continua are chain connected we can add weakly chainedto the list of equivalent conditions.
References [1] N. Brodskiy, J. Dydak, B LaBuz, A. Mitra.
Rips complexes and covers in the uniform cate-gory , Houston J. Math. 39 (2013), no. 2, 667-699.[2] C. Plaut,
Weakly chained spaces , arXiv:2001.03112.
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