# 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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### Is it possible to construct Hausdorff compact topology on every set?

I'd like to know if it's possible to construct Hausdorff compact topology on every set. Assume the axiom of choice if needed. Thanks for ideas.
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### Sum of compact sets is compact without using continuity [duplicate]

I want to show that if $A$ and $B$ are compact sets, then $A+B$ (that is, the set $\{a+b : a \in A , b \in B\}$) is compact. I know that $A+B$ is bounded, but am having trouble showing that it is ...
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### Compactness, why is $(0,1)$ not compact? I need the “thought process” [duplicate]

See, I am told that $(0,1)$ is not compact as a subspace of $\mathbb{R}$. Question is, how do I conclude that? The hint says $(\epsilon,1)_{\epsilon>0}$ does not have a finite subcover. But ...
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### Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
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### Property of compact metric space

Let $X$ be a compact metric space. Let $A$ be a closed subset of $X$ and let $x\in X$ be a point not in $A$. Show that there exists two disjoint open sets $U$ and $V$ such that one contains $A$ and ...
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### Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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### Continuous function on a non-compact set

I'm trying to show if $X$ is non compact ($X \subseteq \mathbb{R}$) then there is a cont function $f:X \rightarrow \mathbb{R}$ which is bounded but doesn't attain it's bounds. I'm trying it for a set ...
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### Why can't a open interval in $\mathbb{R}$ be compact?

If I choose $(0,1)$ and do a covering like this: $(-1,1/2)$ and $(1/3,2)$ it's a finite covering, then $(0,1)$ is a compact set?
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### E is infinite subset of compact set, then is E' also a subset?

Here's a theorem in Rudin's Principles of Mathematical Analysis. 2.37 Theorem: If E is an infinite subset of a compact set K, then E has a limit point in K. Proof: If no point of K were a limit ...
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### Can every compact subset of $\Bbb R^n$ be written as a disjoint union of compact subsets, where each of them are path-connected?

I was wondering if every compact subset of $\Bbb R^n$ could be written as a disjoint union of compact subsets, where each of them are path-connected, i.e. : If $X \subset \Bbb R^n$, $n \ge 1$, $X$ is ...
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### Prove compact set and index set

$$\forall α \in I\; A_α \text{ is a compact set} \implies \bigcap \{ A_\alpha : \alpha \in I\} \text{ is a compact set}$$ How to prove this problem? I know definition of ...
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### If $A$ is compact then $f^{-1}(A)$ compact?

Let $f$ be a continuous function. I know that if $A$ is compact then $f(A)$ is compact but is $f^{-1}(A)$ also compact? I believe it is not but how can I prove it by a counter example?
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### Compact set of closed subset problem

K is subset of complex plane. K is compact set and A is closed subset of K then A is compact set. How to prove this problem?? I know definition of compact set, but i'm not use definition to problem.
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### How to prove intersection compact problem? [duplicate]

If A and B are compact set, then A∩B is compact set. How to prove this problem?. I know compact, but not use to this problem...
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### Is my proof of the fact that the product of two compact metric spaces is compact correct?

Sometimes ago I have posted this question. After sometime of working I think that I have found out a different proof (not "purely topological"). I didn't post it there as an answer because (1) the ...
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### Topological proof of the compactness of product metric space

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric spaces (see the definition here). Then show that the product metric space $(X\times Y,d_{X\times Y})$ is also compact. Now this can be done ...
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### Prove that $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ is not compact

Let $K=\{(x,y,z)\in \Bbb{R}^3\ :\ x^2+yz=x+1\}$ Show that $K$ is not compact
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### Is every compact space hereditarily Lindelöf?

All spaces are assumed Hausdorff. We call a topological space compact if every open cover has a finite subcover. We call it Lindelöf if every open cover has a countable subcover, and hereditarily ...
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### Show that a compact metric space $X$ is locally compact

Assume that $X$ is a compact metric space, that is by definition, every sequence in $X$ has a convergent subsequence. Locally compact means that every point in $X$ has a compact neighbourhood. That ...