Small loop spaces and covering theory of non-homotopically Hausdorff spaces

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of $X$ is the universal covering in categorical sense. Also, we introduce the notion of semi-locally small loop space which is the necessary and sufficient condition for existence of universal cover for non-homotopically hausdorff spaces, equivalently existence of small covering spaces. Also, we prove that for semi-locally small loop spaces, $X$ is a small loop space if and only if every cover of $X$ is trivial if and only if $\pi_1^{top}(X)$ is an indiscrete topological group.Comment: 7 page

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.Comment: 14 pages, 2 figures. arXiv admin note: text overlap with arXiv:1102.0993 by other author

On locally 1-connectedness of quotient spaces and its applications to fundamental groups

Let $X$ be a locally 1-connected metric space and $A_1,A_2,...,A_n$ be connected, locally path connected and compact pairwise disjoint subspaces of $X$. In this paper, we show that the quotient space $X/(A_1,A_2,...,A_n)$ obtained from $X$ by collapsing each of the sets $A_i$'s to a point, is also locally 1-connected. Moreover, we prove that the induced continuous homomorphism of quasitopological fundamental groups is surjective. Finally, we give some applications to find out some properties of the fundamental group of the quotient space $X/(A_1,A_2,...,A_n)$.Comment: 11 page