# 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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### Why is compactness so important?

I've read many times that 'compactness' is such an extremely important and useful concept, though it's still not very apparent why. The only theorems I've seen concerning it are the Heine-Borel ...
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### Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or ...
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### Given a fiber bundle $F\to E\overset{\pi}{\to} B$ such that $F,B$ are compact, is $E$ necessarily compact?

Consider a (locally trivial) fiber bundle $F\to E\overset{\pi}{\to} B$, where $F$ is the fiber, $E$ the total space and $B$ the base space. If $F$ and $B$ are compact, must $E$ be compact? This ...
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### Countable compact spaces as ordinals

I heard at some point (without seeing a proof) that every countable, compact space $X$ is homeomorphic to a countable successor ordinal with the usual order topology. Is this true? Perhaps someone can ...
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### Are compact spaces characterized by “closed maps to Hausdorff spaces”?

It is well known that any continuous map from a compact space to a Hausdorff space must be a closed map. Does this fact characterize compactness? That is, if for a space $X$, every continuous map to ...
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### Compactness in $\mathbb{Q}$

Proving that $[0,1]\subset\mathbb{R}$ is compact you make use of completeness and this is a fundamental step in order to characterize compact subsets of $\mathbb{R}$. Trying to state an analogous ...
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### Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
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### Can compact sets completey determine a topology?

Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then ...
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### Is there a “tree-like” proof of compactness theorem in the case of uncountably many variables?

I like proofs using trees and König's lemma, since they are very visual. One of the applications of König's lemma you can show to students is proving compactness theorem for propositional calculus, ...
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### Are there non-Hausdorff examples of maximal compact topologies in the lattice of topologies on a set?

In the lattice of topologies on a set $X$, the compact topologies are a lower set in the lattice, while the Hausdorff topologies are an upper set. A result of this theorem is that the compact ...
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### Pseudocompactness does not imply compactness

It is well known that compactness implies pseudocompactness; this follows from the Heine–Borel theorem. I know that the converse does not hold, but what is a counterexample? (A pseudocompact space is ...
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### A non-compact topological space where every continuous real map attains max and min

Today I learnt in class that if $X$ is compact then any continuous map $f:X\to\mathbb{R}$ attains max and min. I was thinking if the converse is true: If every continuous map $f:X\to\mathbb{R}$ ...
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### compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
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### Topology: Example of a compact set but its closure not compact

Can anyone gives me an example of a compact subset such that its closure is not compact please? Thank you.
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### Understanding the definition of a compact set

I just need a bit of help clarifying the definition of a compact set. Let's start with the textbook definition: A set $S$ is called compact if, whenever it is covered by a collection of open sets ...
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### Difference between closed, bounded and compact sets

In real analysis, there is a theorem that a bounded sequence has a convergent subsequence. Also, the limit lies in the same set as the elements of the sequence, if the set is closed. Then when ...
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### Metrizable compactifications

Suppose $X$ is a metric space. When does it have a metrizable compactification? Of course it is enough to discuss complete metric spaces, but separability may not be assumed here. I know that ...
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### Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to Y$...
I was wondering if someone can please help me with the following problems: Show that $\mathbb{Q}$ is not locally compact. Prove that if $X$ is Lindelöf and $Y$ is compact then $X \times Y$ is ...