# 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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### Convergence in compact-open topology implies uniform convergence on compacts

Let $X$ be a topological space and $Y$ a metric space. We give $C(X,Y)$ the compact open topology. If $f_n\to f$ in $C(X,Y)$ then $f_n$ converges uniformly to $f$ on every $K\subseteq X$ compact. ...
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### Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
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### Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true? $a.$ If $X$ has at least $n$ distinct points, then the dimension of $C(X)$, the space of continuous real ...
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### Show compactness of subset of $\mathbb R^3$

I need to show that $$A:=\{(x,y,z)\in\mathbb R^3; 3x^3y+2xyz^3+2y^2+3=0, xy^3+3xz+x^3=0\}$$ is closed and bounded, hence compact. I don't really know what to do here, can you help?
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### $X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n$...
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### Examples of compact sets that are infinite dimensional and not bounded

In an infinite dimensional Banach space, does a compact subset have to be finite dimensional? I know it cannot contain any infinite dimensional balls, if this mean it has to be finite dimensional, ...