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

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 $\psp$. By a Spanier space we mean a space $X$ such that $\psp=\pi_1(X,x)$, for every $x\in X$. 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.

💡 Research Summary

The paper investigates the role of Spanier groups in the covering theory of spaces that are not homotopically path Hausdorff. Starting from the notion introduced by Fischer, Repovš, Virk and Zastrow, the authors define the (unbased) Spanier group π_sp¹(X,x) as the intersection of the subgroups π(U,x) obtained from all open covers U of X. A space X is called a Spanier space when π_sp¹(X,x)=π₁(X,x) for every base point; if the equality holds for the based version π_bsp¹, the space is a based Spanier space. The paper first provides an explicit example (the space Y) showing that a Spanier space need not be a based Spanier space, thereby distinguishing the two concepts in the non‑locally path‑connected setting.

The authors then explore how Spanier groups interact with covering maps. For any covering p:Ė→X and any base point x∈X, they prove the chain of inclusions π_bsp¹(X,x) ≤ π_sp¹(X,x) ≤ p_*π₁(Ė,ẋ). Consequently, every loop whose class lies in π_sp¹ lifts to a closed loop in any covering, and a space that fails to be homotopically path Hausdorff cannot possess a simply connected universal covering. This motivates the introduction of “Spanier coverings”: coverings for which the induced subgroup equals the Spanier group, i.e. p_*π₁(Ė,ẋ)=π_sp¹(X,x). Such coverings are shown to be categorical universal objects, and a covering is a based Spanier covering precisely when p_*π₁(Ė,ẋ)=π_bsp¹(X,x).

The central result of the paper is a necessary and sufficient condition for the existence of Spanier coverings. The authors define a space to be semi‑locally Spanier if each point x has an open neighbourhood U such that every loop in U represents an element of π_sp¹(X,x). They prove that a connected, locally path‑connected space admits a Spanier covering if and only if it is semi‑locally Spanier. Moreover, when a Spanier covering exists, for any subgroup H of π₁(X,x) containing π_sp¹(X,x) there is a covering p_H:Ė_H→X with p_{H*}π₁(Ė_H,ẋ)=H. This generalizes the classical correspondence between subgroups and covering spaces, replacing the usual requirement of semi‑local simple connectivity with the weaker semi‑local Spanier condition.

In the final section the paper turns to the topological structure of Spanier groups inside the topological fundamental group π₁^τ(X,x). For connected, locally path‑connected spaces, π_sp¹(X,x) is a closed subgroup of π₁^τ(X,x). Hence, if π₁^τ carries the indiscrete topology, the space must be a Spanier space. When π_sp¹ is trivial, π₁^τ is T₁, and if X is also paracompact, π₁^τ becomes Hausdorff. These observations give concrete criteria for when the topological fundamental group is Hausdorff, linking algebraic properties of Spanier groups with separation axioms.

Overall, the paper extends covering theory beyond homotopically path Hausdorff spaces by introducing Spanier coverings and the semi‑local Spanier condition, provides a clear classification of coverings in this broader context, and connects these ideas to the topology of the fundamental group. The results both generalize classical theorems and supply new tools for studying spaces with pathological local homotopy behavior.