Small loop spaces and covering theory of non-homotopically Hausdorff spaces

Abstract

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of X is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, X is a small loop space if and only if every cover of X is trivial if and only if π top 1 (X) is an indiscrete topological group.2010 Mathematics Subject Classification. 57M10, 57M05, 55Q05, 57M12.

The revised and uniform fundamental groups and universal covers of geodesic spaces

“…Except for the uniform universal cover, we will not need or use any of the recent, non-traditional generalizations of universal covers that relax the evenly covered property (cf. [5], [6], [15], [20]). When we use the uniform universal cover, it will always be explicitly referenced as such, so no confusion should result.…”

“…In particular, π cl (X) contains what Pakdaman, Torabi, and Mashayekhy call in [20] the SG-subgroup. This is the subgroup π sg 1 (X) ⊂ π 1 (X) generated by all small loop lollipops, or path loops of the form αβα −1 , where α is a path beginning at * and β is a small path loop based at α(1).…”

Topological Fundamental Groups and Small Generated Coverings

“…As it is shown in [11], there exist special subgroups of fundamental groups of non-homotopically Hausdorff spaces which have great influence on their coverings. In fact, if a space X is not homotopically Hausdorff, then there exists x ∈ X and a nontrivial loop in X based at x which is homotopic to a loop in every neighborhood U of x (see [8] for the definition of homotopically Hausdorffness).…”

“…In general, various points of X have different small loop groups and hence in order to have a subgroup independent of the base point, Virk [14] introduced the SG (small generated) subgroup, denoted by π sg 1 (X, x), as the subgroup generated by the following set {[α * β * α −1 ] | [β] ∈ π s 1 (X, α(1)), α ∈ P (X, x)}, where P (X, x) is the space of all paths from I into X with initial point x. Virk [14] calls a space X a small loop space if π s 1 (X, x) = π 1 (X, x) = 1 for all x ∈ X. The authors [11] showed that for a covering p : ( X,x) → (X, x) the following relations hold: π s 1 (X, x) ≤ π sg 1 (X, x) ≤ p * π 1 ( X,x). It should be noted that by a result of Spanier [13, §2.5 Lemma 11] one has π sg 1 (X, x) ≤ π(U, x) ≤ p * π 1 ( X,x), ( * )…”

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

“…Also, the authors [10,13] studied the covering theory of non-homotopically Hausdorff spaces. Accordingly, we would like to study coverings of non-homotopically path Hausdorff spaces and investigate the topology type of their fundamental group and their universal covering spaces.…”