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Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $ …

And when can this subspace be chosen to be open? As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds …

In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\s …

Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected. Can we always find an open $V$ such that $K\subset V\subset\overl …

This came up when thinking about this question. It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is …

My first attempt to ask this question sort of failed (I'll explain below). This came up when thinking about this question. It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb …

Let $X$ be a locally connected topological space, which is covered by open sets $\{U_{\alpha},\alpha\in A\}$ and let $C$ be an arc in $X$, i.e. a homeomorphic image of an interval. Is it always possi …

This is an outgrowth of this question. For a (metrizable) space $X$ consider the following increasingly strong properties: (i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\s …

This question is inspired by this one, or rather the way I understood it. Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\mat …

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\varphi …

This is the missing ingredient towards answering my previous question. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). …

The question as stated still does not make much sense, and I think, this is why it does not attract the attention it deserves. Let me restate the problem and tell what I have so far. For a Jordan …