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Coarser Connected Metrizable Topologies Lynne Yengulalp University of Dayton, [email protected]

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Coarser Connected Metrizable Topologies Lynne Yengulalp University of Dayton, Lyengulalp1@Udayton.Edu University of Dayton eCommons Mathematics Faculty Publications Department of Mathematics 9-2010 Coarser Connected Metrizable Topologies Lynne Yengulalp University of Dayton, [email protected] Follow this and additional works at: https://ecommons.udayton.edu/mth_fac_pub Part of the Applied Mathematics Commons, and the Geometry and Topology Commons eCommons Citation Yengulalp, Lynne, "Coarser Connected Metrizable Topologies" (2010). Mathematics Faculty Publications. 40. https://ecommons.udayton.edu/mth_fac_pub/40 This Article is brought to you for free and open access by the Department of Mathematics at eCommons. It has been accepted for inclusion in Mathematics Faculty Publications by an authorized administrator of eCommons. For more information, please contact [email protected], [email protected]. Coarser connected metrizable topologies Lynne Yengulalp1 Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA Abstract We show that every metric space, X , with w(X) ≥ c has a coarser connected metrizable topology. Key words: Coarser connected topology, extent, metrizable 2008 MSC: 54A10, 54A25, 54D05, 54D15, 54D70, 54E35 1. Introduction, Definitions, Notation The study of coarser connected topologies was started by Tkacenko,ˇ Tkachuk, and Uspenskij who developed necessary and sufficient conditions for a topolog- ical space to have coarser connected Hausdorff or regular topology [5]. Con- tinuing the development, Gruenhage, Tkachuk, and Wilson showed that any noncompact metrizable space has a coarser connected Hausdorff topology [4]. Fleissner, Porter and Roitman showed that any zero-dimensional metrizable space with weight at least c has a coarser connected metrizable topology [3]. We improve the last result by removing the zero-dimensional hypothesis. We show that every metric space with weight at least c has a coarser connected metrizable topology. We write (X; τ; µ) for a metric space X with metric µ and corresponding topology τ . For a set A and cardinal λ, we write [A]λ for the collection of sub- sets of A of cardinality λ and [A]<λ for the collection on subsets of cardinality less than λ. The extent, e(X), of a space X is the supremum of cardinals, λ, such that there is a closed discrete subset of X of size λ. If a space, X , has a closed discrete set of size e(X) we say the extent of X is attained in X . Suppose U is a subset of a space X . If C is a closed discrete (with respect to U ) subset of U , then C is not necessarily closed in X . Hence, it is possible that e(U) is attained in U , but there is no closed discrete (in X ) subset of U of 1Department of Mathematics, University of Dayton, Dayton, Ohio 45469, USA [email protected] Preprint submitted to Elsevier February 2, 2010 size e(U). If, on the other hand, there is a closed discrete (in X ) subset of U of size e(U), we say that e(U) is attained in X . For a cardinal κ and positive real number , the hedgehog with spininess κ and spine length is the space Z = f(α; r): α 2 κ, r 2 (0; ]g [ f(0; 0)g with metric ρ defined as follows: r + r0 if α 6= α0 ρ((α; r); (α0; r0)) = jr − r0j if α = α0: The hedgehog space is a connected metric space. 2. Some Lemmas If X is a metric space such that w(X) < c, then X does not necessarily have a coarser connected metrizable topology. For example the disjoint union of countably many Cantor sets is a separable, metrizable, disconnected space with no coarser connected regular topology [5]. We aim to show that all metric spaces, X , with w(X) ≥ c have a coarser connected metrizable topology. This result has already been proven if e(X) is attained (Druzhinina [1], Fleissner, Porter, Roitman [3]). If e(X) = c, then we will see later that e(X) is attained. So, we consider metric spaces, X , such that e(X) > c is not attained. In the proof of the main theorem, we partition the space (X; τ; µ) into sub- sets Wi for i 2 ! such that e(Wi) is attained in X . Lemma 2.1 will be used to define a coarser topology on X in which Wi is connected. We show that µ can be adjusted so that Wi is connected without changing µ on X n Wi . Lemmas 2.2–2.5 discuss the consequences of e(X) not being attained and 0 provide a tool for the partition of X into Wi s. We apply Lemma 2.6 to a particular compact subset of X for the purpose of defining the partition of X 0 into Wi s. Lemma 2.1. Suppose that Y is a subset of a metric space (X; τ; µ), e(clY ) ≥ c is attained in X by C ⊂ int(Y ) and diamµ(clY ) ≤ . If U ⊂ Y is an open set containing C , then there is a coarser topology τ 0 on X generated by a corresponding metric µ0 such that i) µ0 ≤ µ + 2, 0 0 ii) clτ Y = clτ Y and τ clY is connected, and 0 iii) µ (XnU)2 = µ (XnU)2 . Proof. Let e(clY ) = κ. So, C is closed discrete in X and jCj = κ. By replacing C with a subset, we may assume that jY n Cj ≥ κ. Let U = fUc ⊂ U : c 2 Cg be a discrete in X collection of open sets in X such that c 2 2 Uc ⊂ cl(Uc) ⊂ U . For each c 2 C define a continuous function fc : X ! [0; ] such that fc(c) = and fc[X n Uc] = f0g. For each x; y 2 X define ∗ µ (x; y) = Σc2C jfc(x) − fc(y)j + µ(x; y). Because C is closed discrete in ∗ ∗ ∗ X , it is easy to check that µ generates τ , µ (XnU)2 = µ (XnU)2 and µ ≤ µ + 2 . Let (Z; ρ) be a hedgehog space with spininess κ and spine length . Let T = f(α; ): α 2 κg and let S = Z n T . The set T is the set of tips of the spines of the hedgehog which is a closed discrete set of size κ. Let D ⊂ Y n C be a dense subset of Y n C of size d(Y ) = e(Y ) = κ. Define a one-to-one map f : Z ! Y such that f[T ] = D and f[S] = C . For x; y in the image if f , let λ(x; y) = minfµ∗(x; y); ρ(f −1(x); f −1(y))g. For all other x; y 2 X , let λ(x; y) = µ∗(x; y). Define a function µ0 on X × X as follows: 0 µ (x; y) = inffλ(x; x1) + λ(x1; x2) + ··· + λ(xn−1; xn) + λ(xn; y)g where x1; x2; : : : ; xn ranges over all finite sequences (including the empty se- quence) of distinct elements of X n fx; yg. Since ρ and µ∗ satisfy the triangle inequality, in defining µ0 it suffices to consider sums of the form ∗ −1 −1 ∗ µ (x; x1) + ρ(f (x1); f (x2)) + ··· + µ (xn−1; xn) −1 −1 (1) +ρ(f (xn); f (y)) where the sum may start or end with either a ρ term or µ∗ term and the terms of ∗ the sum alternate between ρ and µ . Since x1; x2; : : : ; xn are in the image of f , note that x1; x2; : : : ; xn 2 D [ C . Moreover, if x2 = Y and y 2 Y each sum ∗ 0 ∗ begins with a µ term and x1 2 Y . Hence, µ (x; y) ≥ µ (x; Y ) ≥ µ(x; Y ). Letting y 2 Y vary, we have µ0(x; Y ) ≥ µ(x; Y ). Now we verify that µ0 is a metric. Since µ0 is an infimum over all finite sequences in X , it satisfies the triangle inequality. It is clear that µ0(x; y) = µ0(y; x) and µ0(x; x) = 0 for all x; y 2 X . It remains to show that µ0(x; y) = 0 implies that x = y . Claim 1. Suppose that for a particular sum in the form of (1) and some m < n, λ(xi; xi+1) < for each i 2 fm; m + 1; : : : ; n − 1g. Then, if λ(xm; xm+1) = −1 −1 ρ(f (xm); f (xm+1)) and xm 2 D, then xm+i 2 C for all i odd and xm+i 2 D for all i even in f0; 1; : : : ; n − mg. −1 −1 Proof. Suppose that λ(xm; xm+1) = ρ(f (xm); f (xm+1)) and xm 2 D. Also suppose that λ(xi; xi+1) < for each i 2 fm; m + 1; : : : ; n − 1g. Fix i even such that 0 ≤ i < n − m, and suppose that xm+i 2 D. Since xm+i 2 D, −1 f (xm+i) 2 T . But since the ρ and µ terms of the sum alternate, −1 −1 λ(xm+i; xm+i+1) = ρ(f (xm+i); f (xm+i+1)) < 3 −1 and hence f (xm+i+1) 2 S . By the definition of f , xm+i+1 2 C . Now, ∗ since λ(xm+i+1; xm+i+2) = µ (xm+i+1; xm+i+2) < and xm+i+1 2 C , it must be that xm+i+2 2 D. So, since xm 2 D, by induction, xm+i 2 C for all i odd and xm+i 2 D for all i even in f0; 1; : : : ; n − mg. Claim 2. µ0(x; y) = 0 implies that x = y . Proof. Suppose there were x; y 2 X such that µ0(x; y) = 0 but x 6= y .
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