On small homotopies of loops

Conner, Gregory R.; Meilstrup, M.; Repovš, Dušan; Zastrow, Andreas; Željko, Matjaž

doi:10.1016/j.topol.2008.01.009

Cited by 29 publications

(53 citation statements)

References 11 publications

(33 reference statements)

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“…If X is a metric space and π 1 (X, x 0 ) is residually n-slender, then X admits a generalized universal covering. The authors of [10] show that X is homotopically Hausdorff at x ∈ X whenever X is 1-U V 0 at x. We improve upon this result in Theorem 6.9 below.…”

Section: Generalized Universal Coveringsmentioning

confidence: 84%

“…To characterize the homotopically Hausdorff property, we apply the construction in Remark 2. 10. Figure 2).…”

Section: The Hawaiian Earring As a Test Spacementioning

confidence: 97%

“…Example 6. 10. The Peano continua A and B in [10] (also appearing in [22]) are sometimes referred to as the "sombrero spaces."…”

Section: Generalized Universal Coveringsmentioning

confidence: 99%

“…Remark 2. 10. If (T, t 0 ) is any space, we may add a "whisker" by forming the one-point union T + = (T, t 0 ) ∨ ([0, 1], 1).…”

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confidence: 99%

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Test map characterizations of local properties of fundamental groups

Brazas

Fischer

2018

J. Topol. Anal.

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Local properties of the fundamental group of a path-connected topological space can pose obstructions to the applicability of covering space theory. A generalized covering map is a generalization of the classical notion of covering map defined in terms of unique lifting properties. The existence of generalized covering maps depends entirely on the verification of the unique path lifting property for a standard covering construction. Given any pathconnected metric space X, and a subgroup H ≤ π 1 (X, x 0 ), we characterize the unique path lifting property relative to H in terms of a new closure operator on the π 1 -subgroup lattice that is induced by maps from a fixed "test" domain into X. Using this test map framework, we develop a unified approach to comparing the existence of generalized coverings with a number of related properties.

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Section: Generalized Universal Coveringsmentioning

confidence: 84%

“…To characterize the homotopically Hausdorff property, we apply the construction in Remark 2. 10. Figure 2).…”

Section: The Hawaiian Earring As a Test Spacementioning

confidence: 97%

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Test map characterizations of local properties of fundamental groups

Brazas

Fischer

2018

J. Topol. Anal.

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“…The importance of Spanier groups was pointed out by Conner, Meilstrup, Repovs, Zastrow and Zeljko [5] and by Fischer, Repoves, Virk and Zastrow [6]. In this section, we study some basic properties of Spanier groups and their relations to the covering spaces.…”

Section: Spanier Coveringsmentioning

confidence: 93%

Spanier spaces and the covering theory of non-homotopically path Hausdorff spaces

Pakdaman

Torabi

Mashayekhy³

2013

Georgian Mathematical Journal

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H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced the Spanier group of a based space (X, x) which is denoted by π sp 1 (X, x). By a Spanier space we mean a space X such that π sp 1 (X, x) = π 1 (X, x), for every x ∈ X. In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on covering theory and introduce Spanier coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study the topological properties of Spanier groups and find out a criteria for the Hausdorffness of topological fundamental groups.

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