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

作者：Mashayekhy Behrooz^*; Pakdaman Ali; Torabi Hamid

来源：Georgian Mathematical Journal, 2013, 20(2): 303-317.

DOI：10.1515/gmj-2013-0016

• Covering space
• Spanier group
• Spanier space
• homotopically path Hausdoiffiless
• small loop homotopically Hausdorffness
• shape injectivity

In [Topology Appl. 158 (2011), 397-408], Fischer et al. introduced the Spanier group of a based space (X, x) which is denoted by pi(sp)(1) (X, x). By a Spanier space we mean a space X such that pi(sp)(1)(X, x) = pi(1)(X, x) for every x is an element of X. In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on the 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 establish criteria for the Hausdoiffness of topological fundamental groups.

• 出版日期2013