On semilocally simply connected spaces

TL;DR

This paper explores the concept of semilocally simply connected spaces, constructs examples, proposes a modified Spanier group for better characterization, and discusses implications for generalized covering spaces.

Contribution

It introduces a new space with specific properties, modifies the Spanier group concept, and clarifies the algebraic conditions for generalized covering space existence.

Findings

Constructed a space semilocally simply connected in Spanier's sense with a non-trivial Spanier group

Proposed a modified Spanier group to better characterize semilocal simple connectivity

Identified the weakest algebraic condition for the existence of generalized covering 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.