On semilocally simply connected spaces

Abstract

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces. While the condition "semilocally simply connected" plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition "homotopically Hausdorff" instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.Comment: 17 pages, 8 figure

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

“…This group is called the unbased Spanier group with respect to U, denoted by π(U, x) [11,6]. The following theorem is an interesting result on the above notion.…”

“…But without locally path connectedness, these results fail since there exists a semi-locally simply connected space with nontrivial Spanier groups corresponding to every its open cover. H. Fischer, D. Repovs, Z.Virk, A. Zastrow [6] proposed a modification of Spanier groups so that the corresponding results will be correct for all spaces. In order to do this, they instead of open sets U also considered "pointed open sets", i.e.…”

“…Definition 1.2. ( [6]) Let X be a space, x ∈ X, and V = {(V i , x i )|i ∈ I} be a cover of X by open neighborhood pairs. Then let π * (V, x) be the subgroup of π 1 (X, x) which contains all homotopy classes having representatives of the following type:…”

“…By these inclusions, there exist inverse limits of these Spanier groups, defined via the directed system of all coverings with respect to refinement. H. Fisher et al [6] called them the unbased Spanier group and the based Spanier group of the space X which we denote them by π sp 1 (X, x) and π bsp 1 (X, x), respectively. They also mentioned that these inverse limits are realized by intersections as follows:…”

“…Fischer et al [6] proved that triviality of based Spanier group implies the existence of generalized universal covering. In fact, they proved that for a space X, if π bsp 1 (X, x) = 1, then X is homotopically path Hausdorff which implies the existence of generalized universal covering.…”