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32 questions linked to/from What should be the intuition when working with compactness?
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19 votes
8 answers
8k views

How to understand the compactness in topology space in intuitive way?
Lenik's user avatar
  • 351
14 votes
7 answers
9k views

I've been trying to wrap my head around the concept of compactness and get an intuitive understand of what it is. The definition used in my text book is the finite subcover definition. A subset $K$ ...
Zachary F's user avatar
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12 votes
0 answers
1k views

You're probably already thinking "silly person" but hear me out. Compactness: Every open cover (of a set $A$) has a finite subcover (this means every cover $\{U_\alpha\}_{\alpha\in I}$ where $A\...
Alec Teal's user avatar
  • 5,650
2 votes
0 answers
349 views

I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite ...
Abhishek Shrivastava's user avatar
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2 votes
0 answers
64 views

A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. I want to know what's the motivation for constructing such sets? I know, this ...
SOUL's user avatar
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226 votes
13 answers
32k views

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 ...
FireGarden's user avatar
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75 votes
12 answers
52k views

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familiar to us. Here I want to collect the ways to prove $[a,b]$...
Paul's user avatar
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38 votes
9 answers
46k views

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
QED's user avatar
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32 votes
6 answers
2k views

Ok, buckle up for a rather long question. I've spent a large portion of today learning about compactness, stemming mainly from this wikipedia article about point-set topology. The article mentions ...
D.R.'s user avatar
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16 votes
9 answers
3k views

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} \...
Christopher King's user avatar
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20 votes
3 answers
10k views

Is every subset of a metric space a metric subspace? A simple proof does justify that all are subspaces, still, wanted to know if I missed something.
Jesse P Francis's user avatar
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8 votes
4 answers
1k views

Given a complete normed space $X=(X,\|\cdot\|)$. Every Cauchy sequence converges in it. I am not able to understand why we can't show that every bounded sequence in $X$ will have a convergent ...
ajay pawar's user avatar
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8 votes
4 answers
355 views

I am not asking for properties equivalent to compactness, but for those that better capture the motivation for compactness, i.e. that explain why compactness is talked about so much. The way I see ...
Anguepa's user avatar
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2 votes
2 answers
759 views

For an object $X$ in a category $C$, there is a functor $C(-\,,X)$ from $C^{\mathrm{op}}$ to Set that assigns to each object $Z$ the set $C(Z,X)$ and to each morphism $f: Y \to Z$ the pullback $f^*$ ...
Clemens Bartholdy's user avatar
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6 votes
2 answers
791 views

I recently found this answer by Qiaochu Yuan but I'm not sure what "finiteness" and "discreteness" function are in the context of compactness. I've read What does it mean when a function is finite? ...
Alec Teal's user avatar
  • 5,650

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