Topology and its Applications, ( ISI ), Volume (198), No (1), Year (2016-1) , Pages (22-33)

#### Title : ( On Topological Shape Homotopy Groups )

Authors: tayyebe nasri , fateme ghanei , Behrooz Mashayekhy Fard , Bibi Hanieh Mirebrahimi Paziquee ,

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#### Abstract

In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces $X$, $Y$, $Sh(X,Y)$, defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces, which make them Hausdorff topological groups. We then exhibit an example in which $\check{\pi}_k^{top}$ succeeds in distinguishing the shape type of $X$ and $Y$ while $\check{\pi}_k$ fails, for all $k\in \Bbb{N}$. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity of $\check{\pi}_k^{top}$ with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the $k$th shape group, induced by the $k$th shape loop space and show that it coincides with the above topology.

#### Keywords

Topological shape homotopy group\sep Shape group\sep Topological group\sep Inverse limit.
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@article{paperid:1052799,
author = {Nasri, Tayyebe and Ghanei, Fateme and Mashayekhy Fard, Behrooz and Mirebrahimi Paziquee, Bibi Hanieh},
title = {On Topological Shape Homotopy Groups},
journal = {Topology and its Applications},
year = {2016},
volume = {198},
number = {1},
month = {January},
issn = {0166-8641},
pages = {22--33},
numpages = {11},
keywords = {Topological shape homotopy group\sep Shape group\sep Topological group\sep Inverse limit.},
}

%0 Journal Article
%T On Topological Shape Homotopy Groups
%A Nasri, Tayyebe
%A Ghanei, Fateme
%A Mashayekhy Fard, Behrooz
%A Mirebrahimi Paziquee, Bibi Hanieh
%J Topology and its Applications
%@ 0166-8641
%D 2016