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

Georgian Mathematical Journal (Impact Factor: 0.34). 05/2011; 20(2). DOI: 10.1515/gmj-2013-0016
Source: arXiv

ABSTRACT 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.

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