Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Topological Coarse Shape Homotopy Groups   Uchillo-Ibanez et al. introduced a topology on the sets of shape morphisms
between arbitrary topological spaces in 1999. In this paper, applying a similar
idea, we introduce a topology on the set of coarse shape morphisms $Sh^*(X,Y)$,
for arbitrary topological spaces $X$ and $Y$. In particular, we can consider a
topology on the coarse shape homotopy group of a topological space $(X,x)$,
$Sh^*((S^k,*),(X,x))=\check{\pi}_k^{*}(X,x)$, which makes it a Hausdorff
topological group. Moreover, we study some properties of these topological
coarse shape homotopoy groups such as second countability, movability and in
particullar, we prove that $\check{\pi}_k^{*^{top}}$ preserves finite product
of compact Hausdorff spaces. Also, we show that for a pointed topological space
$(X,x)$, $\check{\pi}_k^{top}(X,x)$ can be embedded in
$\check{\pi}_k^{*^{top}}(X,x)$.

22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex