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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4
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283 views

If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding $Y\hookrightarrow\beta ...
7
votes
1answer
188 views

Clopen subsets of a compact metric space

I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
3
votes
2answers
204 views

$f:X\to X$ is one-one and continuous on a compact space. Is $f$ surjective?

Let $(X,\mathcal T)$ be a compact Hausdorff topological space and $f:X\to X$ be one-to-one and continuous. Is $f$ surjective?
2
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2answers
46 views

Question about finite sets/compactness

I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want ...
1
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1answer
57 views

On compact topological group

Must a compact topological group be metrizable? If not, is it separable? Thanks for any help.
16
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10answers
3k views

How to prove $[a,b]$ is compact?

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 familar to us. Here I want to collect the ways to prove $[a,b]$ ...
3
votes
2answers
83 views

Is every countably compact space feebly compact?

A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite. Every compact space is feebly compact but how about countably compact spaces?
5
votes
1answer
224 views

Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?

It is trivial to see that: If $X$ is Hausdorff, then every compact subset of $X$ is closed. I am asking under what condition does the converse hold, i.e. when does If every compact subset of $X$ is ...
2
votes
2answers
81 views

Is it true that a metric compact is complete space?

Is it true that a metric compact is complete space? I think that it is true.
1
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2answers
127 views

Hausdorff space and Cantor's intersection theorem

$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact. I tried to prove by ...
7
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0answers
107 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
6
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2answers
305 views

one point compactification

I am asked to describe the one point compactification of $(0,1) \cup [2,3)$ of $\Bbb R$ and if I'm not mistaken it is just a circle union the closed set [2,3] correct? Am I missing something?
3
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0answers
46 views

Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
2
votes
1answer
229 views

How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?

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 ...
4
votes
2answers
157 views

Compact space and Hausdorff space

A continuous map from a compact space to a Hausdorff space is closed. Why this is true? Help me please I want to learn why this is correct.
1
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1answer
333 views

Separating disjoint compact sets in Hausdorff space by using open sets

If $X$ is a Hausdorff Space and there are 2 disjoint compact sets $A,B\subset X$, we want to prove $\exists V,U \subset X, V \bigcap U=\emptyset $ S.T. $A\subset U, B\subset V$.($U,V$ are open sets) ...
2
votes
1answer
56 views

A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
1
vote
1answer
106 views

Compactness and connectedness on $M_n(\mathbb R)$

Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices. Which of the following are compact and which are connected? a) The set of all invertible matrices b) The set of all orthogonal ...
2
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2answers
185 views

Compact inclusion in $L^p$

Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$? What is the counterexample if what I said is wrong? Thank you.
8
votes
7answers
726 views

Give an example of a simply ordered set without the least upper bound property.

In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact." (The LUB ...
11
votes
1answer
187 views

Does there exist a topology for a set $X$ which is compact and Hausdorff?

For every set $X$ and every topology $\tau$ over $X$ we have that $\tau$ contains the trivial topology $\{ X, \emptyset\}$, which is compact, and is contained in the discrete topology $\{ S: S ...
1
vote
1answer
60 views

Pointwise Sup of continuous functions using Compactness

Consider two topological spaces $X$, $Y$. Assume $X$ is compact. Question: Is it true that for any continuous function $f:X\times Y\rightarrow Z$ (where $X\times Y$ has the product topology and ...
4
votes
1answer
60 views

Compactification of a discrete space using ultrafilters.

I want to show for the collection of ultra filters on a (non-empty) set $A$, $U(A)$. That $U(A)$ is compact where the topology is derived from the base $U_B = \{F\in U(A)|B\in F\}$. Seeing as $A$ can ...
7
votes
1answer
99 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
1
vote
2answers
124 views

Let $ f:(X, d)\mapsto (X, d ) $ be a mapping on compact metric space with $ d (f (x), f (y))<d (x,y) $for $ x\ne y $

I prove that $ f $ has a fixed point. My question is whether the point is unique and the mapping $ f $ is continuous.
0
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1answer
64 views

Metacompactness of the Euclidean space

Does anyone know how to prove that every Euclidean space is countably metacompact? In particular, my interest is in $R^2$. Thanks,
3
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1answer
76 views

How to show that a continuous map on a compact metric space must fix some non-empty set.

Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$ I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
4
votes
3answers
79 views

Construction of Compactification.

Let $U(S)$ be the collection of ultrafilters on a non empty set $S$. And let $O(A) = \{U\in U(S)| A\in U\}$ for $A\subseteq S$. I am told to show that for $\tau = \{O(A)\subseteq U(S)| A\subseteq ...
4
votes
1answer
283 views

Poincaré inequality and Rellich Theorem in one dimensional weighted Sobolev space

Consider the weighted Sobolev space $W^{1,2}\big((0,R),r^{N-1}\big)$, $N=2,3,\ldots$ and its subspace $W_0^{1,2}\big((0,R),r^{N-1}\big)$. Anyone knows if the Poincaré inequality is true in this case? ...
1
vote
1answer
48 views

Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property

$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $. To prove ...
1
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1answer
52 views

Partition on metric space

Let be $\gamma = \{C_1,\ldots, C_k\}$ a partition of a compact metric space $X$ such that $diam(C_j)<\delta$ for all $j$. Suppose that there exist compact sets $L_i\subset C_i$ for all ...
3
votes
1answer
148 views

Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
3
votes
1answer
169 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
3
votes
1answer
179 views

Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.

I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification. Does this hold in general? I.e given a compact ...
3
votes
1answer
75 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
4
votes
1answer
71 views

Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
0
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1answer
115 views

equivalence of compactness and countably compactness

Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
1
vote
1answer
19 views

compactness - show that there an absolute max

This is a practice question from "Advanced Calculus, Folland. Chapter 1.7 Q.4) Suppose $\quad S\subset { R }^{ n }\quad $ is compact $\quad f:S\longrightarrow R\quad $ is continuous and ...
2
votes
1answer
68 views

exercises in compactness

I am working on some practice problems on Compactness. (Q.1.a Chapter 1.7 in Advanced Calculus, Folland) The question is : Give an example of : a closed set $S\subset R\quad$ and a continuous ...
3
votes
1answer
43 views

No unbounded real continuous function on $X$ can be extended to a continuous real function on $\beta X$

By the Čech-Stone compactification theorem, I know that if $X$ is Tychonoff and $f:X\to [a,b]$ is continuous then $f$ can be extended to $\hat{f}:\beta X\to [a,b]$. How can we show that no unbounded ...
1
vote
1answer
49 views

Show $\tau=\tau^*$ if $\tau^*\subset \tau$ [duplicate]

Let $(X,\tau)$ be compact and $(X,\tau^*)$ be a Hausdorff space. How can we show that $\tau=\tau^*$ if $\tau^*\subset \tau$?
1
vote
1answer
226 views

SHOW that there are infinitely many equivalence classes of formulas

Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the ...
7
votes
2answers
108 views

dense subspace of $\beta \Bbb N \times \beta \Bbb N$

Let $\beta \Bbb N$ be a Čech-Stone compactification of the discrete space $\Bbb N$ and fix a point $p\in \beta \Bbb N\setminus \Bbb N$. Put $X=\Bbb N\cup \{p\}$ and $Y=\beta \Bbb N \setminus \{p\}$. I ...
6
votes
4answers
2k views

A example of closed and bounded does not imply compactnesss in metric Space

Let $X$ be the integers with metric $ρ(m,n)=1$, except that $ρ(n,n)=0$. Check that $ρ$ is a metric. Show that $X$ is closed and bounded, but not compact. This is a "made-up" example demonstrating ...
4
votes
1answer
110 views

Question about Čech-Stone compactification

Let $\beta X$ be the Čech-Stone compactification of $X$ and $p\in \beta X\setminus X$. Is it true that $\{p\}$ can not be a $G_\delta$ set ?
0
votes
2answers
34 views

Prove that $D ⊂\Bbb R^{n}$ is compact iff whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$

Prove that $D ⊂\Bbb R^{n}$ is compact if and only if whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$ , there is a finite subcollection satisfying ...
2
votes
3answers
241 views

Norm equivalence of a vector norm and its induced matrix norm using compactness argument

I have a theorem in my book on matrix computations that states the following: A vector norm and its induced matrix norm satisfy the inequality: $\|Ax\|\leq \|A\|$$\|x\|$ where A $\in R^{nxn}$ and x ...
5
votes
1answer
552 views

closed bounded subset in metric space not compact

Let $\ell^{\infty}$ be the space of bounded sequences of real numbers, endowed with the norm $\|\mathbf x\|_\infty=\sup_{n\in N}|x_n|$, where $\mathbf x=(x_n)_{n\in\Bbb N}$. Prove that the closed ...
7
votes
1answer
128 views

Stone-Čech compactification of completely regular space

Suppose $X$ is a completely regular space. Let $M$ be the set of nonzero algebra homomorphisms from $BC(X,\mathbb{R})$ to $\mathbb{R}$, equipped with the topology of pointwise convergence. Show that ...
2
votes
0answers
125 views

For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$

The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...