arXiv subjects
Publications and source records attributed to Tayyebe Nasri.
In 2002, Biss investigated on a kind of fibration which is called rigid covering fibration (we rename it by rigid fibration) with properties similar to covering spaces. In this paper, we obtain a relation between arbitrary topological spaces and its rigid fibrations. Using this relation we obtain a commutative diagram of homotopy groups and quasitopological homotopy groups and deduce some results in this field.
In this paper, we introduce a subcategory $\widetilde{Sh}_*$ of Sh$_*$ and obtain some results in this subcategory. First we show that there is a natural bijection $Sh (Σ(X, x), (Y,y))\cong Sh((X,x),Sh((I, \dot{I}),(Y,y)))$, for every $(Y,y)\in \widetilde{Sh}_*$ and $(X,x)\in Sh_*$. By this fact, we prove that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_*$, $\checkπ_n^{top}(X,x)\cong \checkπ_{n-k}^{top}(Sh((S^k, *),(X,x)), e_x)$, for all $1\leq k \leq n-1$.
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 $(X,x)$, $Sh^*((S^k,*),(X,x))=\checkπ_k^{*}(X,x)$, 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 $\checkπ_k^{*^{top}}$ preserves finite product of compact Hausdorff spaces. Also, we show that for a pointed topological space $(X,x)$, $\checkπ_k^{top}(X,x)$ can be embedded in $\checkπ_k^{*^{top}}(X,x)$.
The paper is devoted to the study of coarse shape of Cartesian products of topological spaces. If the Cartesian product of two spaces $X$ and $Y$ admits an HPol-expansion, which is the Cartesian product of HPol-expansions of these spaces, then $X\times Y$ is a product in the coarse shape category. As a consequence, the Cartesian product of two compact Hausdorff spaces is a product in the coarse shape category. Finally, we show that the shape groups and the coarse shape groups commute with products under some conditions.
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 $\checkπ_k^{top}$ succeeds in distinguishing the shape type of $X$ and $Y$ while $\checkπ_k$ fails, for all $k\in \Bbb{N}$. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity of $\checkπ_k^{top}$ 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.