Topology and its Applications, ( ISI ), Volume (232), No (1), Year (2017-12) , Pages (242-255)

#### Title : ( Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups )

Authors: seyed zeynal pashaei , Behrooz Mashayekhy Fard , Hamid Torabi Ardakani , Mehdi Abdullahi Rashid ,

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#### Abstract

Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$. In this paper, we extend the concept of $X$ being SLT space to $H$-SLT space at $x_0$. First, we show that the fibers of the endpoint projection $p_{H}:\widetilde {X}_{H}\rightarrow X$ are topological group when $X$ is $H$-SLT space at $x_0$ and $H$ is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to $H$ and homotopically Hausdorff relative to $H$ coincide. Moreover, among other things, we show that the endpoint projection map $p_{H}$ has the unique path lifting property if and only if $H$ is a closed normal subgroup of $\pi_{1}^{qtop}(X,x_{0})$ when $X$ is $H$-SLT at $x_{0}$. Second, we present conditions under which the whisker topology agrees with the quotient of compact-open topology on $\widetilde{X}_{H}$. Also, we study the relationship between open subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{qtop}(X,x_{0})$.

#### Keywords

, Small loop transfer space, quasitopological fundamental group, whisker topology, homotopically Hausdorff, homotopically path Hausdorff, covering map, semicovering.
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@article{paperid:1064875,
author = {Pashaei, Seyed Zeynal and Mashayekhy Fard, Behrooz and Torabi Ardakani, Hamid and Abdullahi Rashid, Mehdi},
title = {Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups},
journal = {Topology and its Applications},
year = {2017},
volume = {232},
number = {1},
month = {December},
issn = {0166-8641},
pages = {242--255},
numpages = {13},
keywords = {Small loop transfer space; quasitopological fundamental group; whisker topology; homotopically Hausdorff; homotopically path Hausdorff; covering map; semicovering.},
}

%0 Journal Article
%T Small Loop Transfer Spaces with Respect to Subgroups of Fundamental Groups
%A Pashaei, Seyed Zeynal
%A Mashayekhy Fard, Behrooz
%A Torabi Ardakani, Hamid
%A Abdullahi Rashid, Mehdi
%J Topology and its Applications
%@ 0166-8641
%D 2017