# Tagged Questions

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### A Theorem About Compactness and

My first exposure to any sort of topology is from Spivak's Calculus on Manifolds. I think I understand compactness conceptually, I'm just finding the rigor a little bit elusive. My first question ...
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### Convergence of compact sets continuous function

Let $X$ and $Y$ be compact set (both subset of the real number). Consider the continuous function $f:X \rightarrow Y$. For any given $y$, and for $h>0$ small enough so that $y+h \in Y$. I want to ...
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### An Open Cover $\mathscr{F}$ of $2\mathbb{N}$ That Has No Finite Subcover

What is an open cover $\mathscr{F}$ for the set $2\mathbb{N}=\{2n:n\in\mathbb{N}\}$ that has no finite subcover? My initial answer is ...
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### Prove the following set is compact

$\def\R{\mathbb R}$Fix vectors $b\in\R^k_+$ and $D\in\R^k_{++}$, and a matrix $A\in\R^{N\times k}$. Here, $\R^k_+$ denotes the set of vectors in $\R^k$ whose entries are nonnegative, and $\R^k_{++}$ ...
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### $d(f,g)=\sup\{\lvert f(x)-g(x)\rvert : x\in[0,1]\}$ [closed]

Let $F$ be the non-empty set of functions that map from $[0,1]$ to $[0,1]$. Is $$d(f,g)=\sup\{\lvert f(x)-g(x)\rvert : x\in[0,1]\}$$ a metric on $F$?
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### $S$ is a compact subset of $\mathbb{R}$ and $T$ is a closed subset of $S$ $\implies$ $T$ compact [duplicate]

If $S$ is a compact subset of $\mathbb{R}$ and $T$ is a closed subset of $S$, then $T$ is compact. How can I show this using the definition of compactness, and separately showing this by the ...
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### “Collection”: What does it mean?

I've seen a lot of question of same ilk as the request I'm about to pose, but what I'd like to know is what does "any collection" mean in the following request: Prove that the intersection of any ...
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### Is this space countably compact

Let $X$ be a Tychonoff countably compact space and $A$ is a subapce of $X$ such that for any countable $B \subset A$ we have $\overline{B} \subset A$. My question is this: Is this subspace $A$ ...
Given convergent sequences of compact sets $\{A_k\}$ and $\{B_k\}$ with $\lim_{k \rightarrow \infty} A_k = A_{\infty}$ and $\lim_{k \rightarrow \infty} B_k = B_{\infty}$, $A_k \cap B_k \neq \emptyset ... 1answer 106 views ### Minimal Connected Set containing a Closed Connected Set in a Compact Space This question came from Dugundji's$\textit{Topology}$: Given a compact, connected space$X$, let$A \subset X$be closed. Prove that there exists a closed, connected set$B \subset X$such that$A ...
We know that if $A$ and $B$ are compact (assuming A and B are non-empty), then the Cartesian product $A \text{x} B$ is compact. But how do you go the other way round. We have to show that any ...