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

Georgian Mathematical Journal (Impact Factor: 0.45). 05/2011; 20(2). DOI: 10.1515/gmj-2013-0016
Source: arXiv


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.

Download full-text


Available from: Behrooz Mashayekhy, Oct 08, 2015
  • Source
    • "X, ˜ x) = π sp 1 ( X, ˜ x) (see [5] "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we study necessary and sufficient conditions for existence of categorical universal coverings using open covers of a given space $X$. In fact, we define several homotopy theoretic conditions which we then prove are equivalent to the existence of a categorical universal covering space. As an application, we show that all universal coverings of a connected and locally path connected space are Spanier spaces.
  • Source
    • "H. Fischer et all [8] called such inverse limit the Spanier group of the based space (X, x) and denoted it by π sp 1 (X, x). The authors [12] introduced Spanier spaces which are spaces such that thier Spanier groups is equal to their fundamental groups. "
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we show that the image of the topological fundamental group of a given space $X$ is dense in the topological fundamental group of the quotient space $X/A$ under the induced homomorphism of the quotient map, where $A$ is a suitable subspace of $X$ with some conditions on $X$. Also, we give some application to find out some properties for $\pi_1^{top}(X/A,*)$. In particular, we give some condition in which $\pi_1^{top}(X/A,*)$ is an indiscrete topological group.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The paper is devoted to show that topological homotopy groups commute with inverse limits under certain circumstances. As a consequence, we present some conditions under which the topological homotopy group of an inverse limit space is a topological group. We also give some conditions for countability of homotopy groups.
Show more