Let $f : X \rightarrow Y$ be a perfect map: a continuous surjective closed map such that $f^{−1}(\{y\})$ is compact for all $y \in Y$ . Prove that if $Y$ is compact, then $X$ is compact. How to use compactness on $Y$ to show it on $X$. Thank you for any help.

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ is continuous, then $\forall \epsilon >0 \ \exists \delta > 0$ such that $$d(x,y) < \delta \implies d(g(x), g(y)) < […]

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence. Let $U$ be a subset of $\mathbb{R}^n$ such that $U$ is not closed. Construct a […]

Is there anything in the Bolzano-Weierstrass proof (for points in $\mathbb{R}^k$) that is analogous to using equicontinuity for functions in Arzela-Ascoli? Does the statement: “points are always equicontinuous” make sense? I feel that I may be forcing this reinterpretation, but I also feel that there might be some connection between the topology of points and […]

A topological space $X$ is called sequentially compact if every sequence of points in $X$ has a subsequence that converges to a point in $X$. I know it’s very similar to Bolzano–Weierstrass theorem but I don’t really know how to argue about it. Any good idea? Thanks for helping 🙂

I am wondering when it is known that a set $A$ in topological space $X$ can be exhausted by compact sets, that is there exists increasing sequence of compact sets covering $A$. I guess this should involve both conditions on topology of $X$ and the set $A$ itself. I remember my lecturer in analysis using […]

Prove if $S_1,S_2$ are compact, then their union $S_1\cup S_2$ is compact as well. The attempt at a proof: Since $S_1$ and $S_2$ are compact, every open cover contains a finite subcover. Let the open cover of $S_1$ and $S_2$ be $\mathscr{F}_1$ and $\mathscr{F}_2$, and let the finite subcover of $\mathscr{F}_1$ and $\mathscr{F}_2$ be $\mathscr{G}_1$ […]

I know that in a metric space $X$ compactness, countable compactness and sequential compactness of a subspace $X’$ are equivalent using the definition of countable compactness as every infinite subset of $X’$ has an accumulation point in $X’$ and of sequential compactness as every sequence in $X’$ has a subsequence converging to a point in […]

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and $x\in X\setminus E$, $\exists f\in\mathscr F$ such that $f(x)\notin\operatorname{cl}[f(E)]$), then $X$ can be densely embedded into a compact Hausdorff space. Namely, considering the compact […]

The question is exactly the title. Is there a good classification of which functions from $\mathbb{N}$ to $\mathbb{N}$ (or, more generally, from $\mathbb{N}^n$ to $\mathbb{N}$)? Also, what is a good source to learn about $\beta\mathbb{N}$? I think I can prove that every function extends uniquely, but this seems a little strong, especially since I can’t […]

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