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

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

2011-05-20

arxiv.org/abs/1105.4010v1

Mathematics

Algebraic Topology

13 pages, 2 figures

Scientific paper

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 some examples 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 prove that the topological fundamental groups of spaces with trivial Spanier groups have $T_1$ topology.

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