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Let $\mathbb{D}=\{z\in \mathbb{R}^2\mid |z|<1\}$ Is it true to say that every homeomorphism of $\mathbb{D}$ is conjugate to a self homeomrphism of the disk extendable to a homeomorphism of $\bar{\math …

About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the deta …

Let $S$ be the suspension operator. Let $K1$ and $K_2$ be two knots in $S^3$ which are not equivalent. Does this imply that their suspensions in 4 sphere are not equivalent in the sense …

Inspired by this question we ask the following question. Note that the comment conversation and answers to the above question imply that There are two complementary subsets of the unit …

Let $A\subset \ell^2$ consist of all $x\in \ell^2$ with $|x|_2=1$ which does not belong to any $\ell^p$ for all $0<p<2$. Note that $A$ is non-empty with a Baire category argument. I am …

Let $X$ be a compact Hausdorff topological space put $A=C(X)$ the $C^*$ algebra of all complex valued continuous functions. The Gelfand correspondence between the category of compact …

By a potential space filling curve we mean a continuous function $f:[0,1]\to [0,1]$ such that there is a continuous surgective function $g:[0,1]\to [0,1]^2$ with $f=\pi_1 \circ g$ where $\pi_1 …

What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space? The Motivation: The motivation comes from the idempotent conjecture of Kapla …

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ there …

In this question we consider the standard inclusion of $\mathbb{C}P^{n}\subset \mathbb{C}P^{n+1}$. It is well known that $\mathbb{C}P^n$ is not a retract of $\mathbb{C}P^{n+1}$. What about if we …

We define an embedding of the set of prim numbers into the Cantor set as follows: First we recall that the cantor set $\mathcal{C}$ is homeomorphic to $(\mathbb{Z}/10\mathbb{Z})^\omega $ since the l …

Let $M,N$ be two disjoint closed holomorphic submanifolds of $\mathbb{C}^n$. Is there a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $f(M)=0,\;f(N)=1$.

Let $X$ be a compact Haussdorf topological space with the following property: For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a path connected subset of $\mathbb{C}$. Is suc …

The First question Let $A$ be a Banach or a $C^*$ algebra. Assume that $e,f$ are two idempotents or prjections in $A$ which satisfy $ef=fe=0$. Assume that there are two automorphisms $\phi, \psi: A \t …

Assume that $f,g:S^{2n}\to \mathbb{R}^{n}$ are $2$ maps. Assume that the set valued map $p(x)=\{f(x),g(x)\}$ is a continuous set valued map. Does there exist a point $p\in S^{2n}$ such that $p(x)=p( …