2012
DOI: 10.4064/fm218-1-2
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Covering maps for locally path-connected spaces

Abstract: Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces.Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connecte… Show more

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Cited by 38 publications
(60 citation statements)
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“…(X, x 0 ). By Theorem 2.1 of [15], There is an open cover U in X such that π(U, x 0 ) ≤ H. On the other hand, by [6,Proposition 4.4], X wh H = X l H . So, using Remark 3.1, we can conclude that for each path δ from x 0 to x, X l [δ −1 Hδ] = X wh [δ −1 Hδ] .…”
Section: Relationship Between Open Subsets Of π Whmentioning
confidence: 99%
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“…(X, x 0 ). By Theorem 2.1 of [15], There is an open cover U in X such that π(U, x 0 ) ≤ H. On the other hand, by [6,Proposition 4.4], X wh H = X l H . So, using Remark 3.1, we can conclude that for each path δ from x 0 to x, X l [δ −1 Hδ] = X wh [δ −1 Hδ] .…”
Section: Relationship Between Open Subsets Of π Whmentioning
confidence: 99%
“…We denote the space X H equipped with this topology by X top H . The second topology is the whisker topology which was introduced by Spanier [14,Theorem 2.5.13] and named by Brodskiy et al [6], as follows. The Spanier group π(U, x) [14] with respect to an open cover U = {U i | i ∈ I} is defined to be the subgroup of π 1 (X, x) which contains all homotopy classes having representatives of the type n j=1 α j β j α −1 j , where α j 's are arbitrary paths starting at x and each β j is a loop inside one of the open sets U j ∈ U.…”
Section: Introductionmentioning
confidence: 99%
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“…Indeed, many authors have studied unusual and pathological examples of covering spaces and tried to buid a suitable general theory. One should mention in particular Fox's theory of overlays [5], an interesting generalization of the concept of coverings by Fischer and Zastrow [6], and a theory of coverings specially geared toward locally path-connected spaces by Brodsky et al [1]. See also Dydak's short note [4] that was written very much in the spirit of the present article.…”
Section: Introductionmentioning
confidence: 98%
“…Generalizations of the notion of covering map and extensions of covering theoretic techniques to spaces beyond those in the classical theory have appeared in many different contexts [2,5,13,14,17,22]. In this paper, we define the notion of a semicovering map and construct a topologically enriched fundamental groupoid functor π τ intimately related to universal constructions of topological groups.…”
Section: Introductionmentioning
confidence: 99%