# Tagged Questions

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness.

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### Motivation of paracompactness

"A paracompact space is a topological space in which every open cover admits a locally finite open refinement" is the definition of paracompactness on Wikipedia. Comparing with the definition of ...
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### One-point compactification of a locally connected space.

Is the one-point compactification of a connected and locally connected space also locally connected? My guess is no, because I haven't been able to prove it. But of course I also haven't come up with ...
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### Points in the boundary of a compact set $K\subset\mathbb{R}^2$ reachable by a path in $K^c$

Let $K\subset\mathbb{R}^2$ be compact. Let the path boundary of $K$ denote the set of points in $z\in K$ such that for some point $w\in K^c$, there is a continuous path $\gamma:[0,1]\to\mathbb{R}^2$ ...
I have been trying to solve the following problem: Let $M \subset \mathbb R^3$ be the set of points $(x,y,z) \in \mathbb R^3$ at which $xy + xz + yz = 1.$ Prove that $M$ is a $2$-dimensional manifold. ...