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Theorem. For a topological space $X$ the following conditions are equivalent: 1) for any continuous map $f:X\times S^2\to\mathbb R^2$ there exists a point $s\in S^2$ such that $f(x,s)=f(x,-s)$ …

If a topological group is contractible, then it is locally contractible (using the group operation produce a contraction which does not move the unit of the group). By a classical result of [A. Gleaso …

The following theorem and example give (partial) answers. By an $n$-manifold we understand a Hausdorff space whose any point has a neighborhood, homeomorphic to an open subspace of $\mathbb R^n_+=\{(x …

The space $X$ constructed in Theorem 1.5 of this preprint has the required properties. This space contains a non-closed compact metrizable subspace $K$, so is not weakly Hausdorff. On the other hand, …

A suitable counterexample can be constructed as follows. Let $\mathbb I=\{x\in\mathbb R:0<x<1\}$ denote the open unit interval and let $B=\{0\}\cup(\omega\times\mathbb I)\cup\{1\}$ be endowed with th …

It is nice that you asked a question about the space $\mathbb Q P^\infty$. I have thought about this space for a long time and came to the conclusion that $\mathbb Q P^\infty$ is the most "regular" sp …

It seems that the answer to this problem is affirmative. We can argue as follows. Assume that $A\subset\mathbb R^2$ is a subspace whose symmetric square $A^2/_\sim$ is homeomorphic to the Mobius stri …

Yes, this is true because every topological copy of $[0,1]$ in the plane is unknotted and can be transformed by a homeomorphism of the plane into the straight arc $[0,1]\times\{0\}$. In the latter cas …

There is no upper bound for dimension in this case. Just consider any surjective continuous map $f:[0,1]\to [0,1]^\omega$ onto the Hilbert cube. Such a map exists since the Hilbert cube is a Peano co …

Let us show how to find such a retraction for $n=2$ (I do not know if this method generalizes to higher dimensions). Given a compact set $C\subset\mathbb R^2$ and an open neighborhood $U\subseteq\mat …

I noticed that this question still has no accepted answer and all existing answers are rather long. It seems that the answer can be easily obtained using some results of infinite-dimensional topology, …

This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of Bessaga and Pelczynski, Chigogidze or Sakai. Factor Theorem. For any Polish abs …

This question concerns manifolds modeled on the direct limit $\mathbb R^\infty$ of Euclidean spaces. The theory of such manifolds is well-developed. The most important results of this theory were obta …

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that pr …

The answer is ``yes'' if $X$ is Hausdorff. We can identify the arc $C$ with the unit interval $[0,1]$ and assume that $[0,1]$ is contained in $X$ and is covered by a family $\mathcal U$ of open subs …