# Questions tagged [compactness]

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. This includes logical compactness.

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### Why can't I immerse the unit interval $[0, 1]$ in the 1-torus $T^1$?

I'm supposed to prove that I can't smoothly immerse a compact, simply connected and smooth 1-manifold into $S^1 = T^1$, the 1-torus. I don't understand why I can't do this: take my simply connected 1-...
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### Proof regarding the equivalence of norms, compactness and completeness of normed vector spaces of finite dimension

I wrote a proof regarding the important properties of normed vector spaces that are of finite dimension. I would like to know if I've made mistakes of any sort. I'm open to any critique. $\mathbb{K}$ ...
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### Quasicompactness of $\operatorname{Proj}A$

$\newcommand{\Proj}{\operatorname{Proj}}$ Let $A$ be a graded ring of the form: \begin{align} A=\bigoplus_{n=0}^\infty A_n \end{align} So in particular this ring is positively graded, but it is not ...
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### What fields generate compact projective spaces?

Given a field $k$, one can generate the vector space $k^n$ and topologize it with the product topology. Then one can delete the 0 element and quotient the remaining set ($k^n \setminus \{0\}$) by the ...
• 295
1 vote
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### Compactness of set of all measurable functions

Let $X$ be compact subset of $\mathbb{R}^n$ and let $Y$ be compact subset of $\mathbb{R}^p$. Consider $\mathcal{F}$ the set of all measurable functions from $X$ to $Y$. Can I find topologies under ...
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### Showing One-point compactification topology on $\mathbb R^n$ is Compact without using Stereographic Projection map. [duplicate]

The following is a well known result from point-set topology: Theorem: Suppose $\mathbb R^n$ is the Euclidean-$n$-space and $\infty$ be a symbol not contained in $\mathbb R^n$(By Russel's Paradox)....
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1 vote
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### Proving $\sigma$-compactness of Pontryagin dual

I am reading A First Course in Harmonic Analysis(2nd ed.) by Anton Deitmar. In Theorem 7.1.4, the author proves that the dual group $\hat{A}$ of a LCA group $A$ is also an LCA group. I am stuck with ...
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### Is there a continuous bijective mapping of $R$ into a compactum

I wonder if there are generally no bijective, continuous mappings of the form $$f: \mathbb{R} \rightarrow K$$ if $K$ is a topological compact space. My considerations I realize that the inverse ...
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### When must a space generated by compacts also be generated by Hausdorff compacts?

Cross-posted to MathOverflow: https://mathoverflow.net/questions/475934/. I'm interested in comparing $k_1$-spaces, spaces whose topologies are witnessed by their compact subspaces, and $k_3$-spaces, ...
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### For compact Hausdorff spaces, is countable pseudocharacter equivalent to first countable? [duplicate]

Let $X$ be a compact $T_2$ space. Is $X$ first countable if, and only if, $X$ has countable pseudocharacter? Note: I have already proven that every $T_1$ first countable space has countable ...
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