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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, …

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 …

Good question. 100 years ago it was much easier to answer it. For example, the "map" of mathematics drawn by Janiszewski in 1915 in the book "Poradnik dla samoukow" looked as follows: Now everything b …

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 …

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 …

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 …

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 $f(x,s)=f(x,-s)$ …

No solenoid can be embedded into the Mobius strip. To derive a contradiction, assume that some solenoid $S$ embeds into the Mobius strip $M$. Let $\pi:C\to M$ be a 2-fold covering map of the cylinder …

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 …

I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ea …

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary). I need a reference to the following facts (which I believe are true at least in dimension $n=3$): Fact 1. For every close …

Definition. A closed subset $S$ of a topological space $X$ is called $\bullet$ a separator of $X$ if $X\setminus S$ is disconnected; $\bullet$ an irreducible separator if $S$ is a separator of $X$ …

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 …

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 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 …