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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3
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190 views

Closed Compact Subset of Product Space Must Have Empty Interior

Suppose that $\{X_{\alpha}\}_{\alpha\in A}$ is a nonempty family of topological spaces. Suppose also that there is an infinite index set $B\subseteq A$ such that $X_{\alpha}$ is not compact for any ...
2
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2answers
2k views

Cartesian product of compact sets is compact

Prove that if two sets $A$ and $B$ are compact then so is their Cartesian product $A \times B = \{(a,b): a \in A, b\in B\}$. The hint is to use Bolzano Weiertrass theorem and an argument of sequence ...
2
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1answer
191 views

The Arzelà–Ascoli theorem fails on a half-open interval

Can we find an example: (1) $\lbrace f_n \rbrace_n$ is a family of real-valued functions defined on $[0,1)$ such that this family is uniformly bounded and equicontinuous, $f_n(0)=0$; ~~~ Uniformly ...
2
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3answers
390 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
2
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1answer
306 views

Compact and Connected.

I have a proof which needs to check by other person. Show: Every nonempty compact connected set in R is of the form [a, b]. My proof: Let's take any nonempty set D in R so there exist a, b, z in D ...
0
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2answers
69 views

Is the boundary of this set compact?

Let $X$ be a topological space, $Y$ a subspace of $X$ and $A\subseteq Y $ such that $\partial(A)$ is compact in $X$. Is $\partial(A)$ compact in $Y$?
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0answers
77 views

prove the compact set with open cover

question is show that (0,1] is not compact by constructing an open cover of (0,1] that does not have a finite sub-cover can we choose the open cover Qn = (1/n,1],n=2,3..... so ( i choose ...
1
vote
2answers
160 views

Show that $SO(3)$ is compact.

In the following we identify $n \times n$ matrices with the space $R^{n^2}$ by sending entries in the first column to the first $n$ coordinates, those in the 2nd column to next $n$ coordinates, and so ...
2
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1answer
79 views

Is this a totally bounded set in the space of continuous functions?

If $A=\{f\in C[0, 1]: \int^1_0|f(x)|^2\,dx\leq1\}$ and metric $d(f, g)$ is $(\int^1_0|f(x)-g(x)|^2dx)^\frac{1}{2}$. Is $A$ totally bounded? I know $A$ is clearly bounded since $d(f, 0)\leq 1$ under ...
0
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1answer
26 views

Help Understanding: Closed Subspace of Compact Space is Compact on ProofWiki

http://www.proofwiki.org/wiki/Closed_Subspace_of_Compact_Space_is_Compact ProofWiki provides the following proof that a closed subspace of a compact space is compact: Let $T$ be a compact space. Let ...
0
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1answer
917 views

Proving that the Intersection of any Collection of Compact Sets is Compact

I was trying to do this problem this way: Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection ...
2
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1answer
68 views

Prove or disprove a set $F$ is closed.

This is an example in my book that talks about $F$ being precompact; Let $F$ be the subset of $C([0,1])$ that consists of functions $f$ of the form $$f(x) = \sum_{n=1}^{\infty}a_n\sin(n\pi x) ...
3
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3answers
175 views

Infinite compact subset of $\mathbb{Q}$

Can I find an infinite set in $(\mathbb{Q},\mathcal{T}_e|_\mathbb{Q})$ which is compact?
2
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2answers
117 views

Compact image is compact?

Let $f: X \to Y$ be a function between metric spaces $X, \phi$ and $Y, d$ such that $f^{-1}(U)$ is open in $X$ for every subset $U$ open in $Y$. Prove that if $C$ is a compact subset of $X$, then ...
2
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3answers
199 views

Compact but not Hausdorff space

I think this space might be a space which is compact and non-Hausdorff, but I don't how to prove this. Let $x,y\in\mathbb{R}^n$, define $x\sim y\leftrightarrow\exists t\neq 0(x=ty)$. Then $\sim$ is ...
4
votes
1answer
79 views

Regarding compactness of a space

I am trying to solve the following problem: Let $X$ be a metrizable topological space. Prove that the following statements are equivalent. (a) $X$ is compact (b) $X$ is bounded with respect to ...
2
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2answers
253 views

Proof that a limit point compact metric space is compact.

If $(X,d)$ is limit point compact, show it is compact. I have found multiple proofs of this that first show that limit point compact implies sequential compact, which in turn implies compactness. I ...
3
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2answers
326 views

Can $X$ have compact connected components?

Let $X$ be a Hausdorff, locally compact but non-compact topological space. If the (Alexandroff) one-point compactification is connected, can $X$ have compact connected components?
0
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1answer
155 views

If the distance between any two points is less than $1$, must $X$ be compact?

Let $X$ be a complete metric space such that the distance between any two points is less than $1$. Then is $X$ necessarily compact? Thanks in advance.
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5answers
1k views

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.
2
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1answer
325 views

How to show pre-compactness in Holder space?

Let $K \in \mathbb{R}^d$ be a compact set and consider the space of Hölder continuous functions $C^{0,\gamma}(K)$ with norm $||f||_{C^{0,\gamma}}:=||f||_{\infty}+\sup_{x,y \in K,x \neq ...
2
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0answers
124 views

Is this proof correct about compact sets inside open sets?

I've been solving the following problem: "If $U\subset\mathbb{R}^n$ is open and $C\subset U$ is compact, show that there is a compact set $D$ such that $C\subset \operatorname{int}(D)$ and $D\subset ...
0
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2answers
74 views

Prove that a continuous $f$ in $(0,1)$ can be extended into its one-point compactification if the limit at both end point exist and equal

Let $X = (0,1)$. Consider the one-point compactification of $X$ (which is homeomorphism to $S^{1}$). Prove that a bounded continuous function $f:(0,1) \rightarrow R$ is extendable to this ...
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1answer
50 views

About the properties of Measure of noncompactness

Let $\alpha$ denote the Kuratowski measure of noncompactness defined on the Banach space $(E,\|.\|)$ and $A, B\subset E$ be nonempty, bounded subsets. Then, how to prove that if $A\subset B$ then ...
3
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2answers
149 views

What is compactification generally?

In wikipedia, compactification is defined as an topological imbedding $f:X\rightarrow Y$ such that $f(X)$ is dense in $Y$. However, Munkres-Topology requires $Y$ to be Hausdorff to be called a ...
2
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1answer
60 views

Compactness of closure

Let $A$ be a subset of a metric space $X$. Assume that each sequence in $A$ has a convergent subsequence with limit in the closure $\overline{A}$. Does this imply that $\overline{A}$ is compact? I ...
3
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0answers
114 views

Smash product of compact spaces

In the topology book I'm reading I found the following statement: The "smash product" (of two pointed spaces) is defined as $X \bigwedge Y=X \times Y/(X \times \lbrace*\rbrace \bigcup Y \times ...
0
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1answer
345 views

In compact-open topology, $C(X,Y)$ is Hausdorff if $Y$ is Hausdorff

Show that in the compact-open topology, $C(X,Y)$ is Hausdorff if $Y$ is Hausdorff and regular if $Y$ is regular In the first statement, let $f,g$ be 2 functions in $C(X,Y)$, we need to ...
4
votes
2answers
299 views

Limit point compactness implies sequential compactness

I am trying to go through the proof of: Suppose $ X $ is metrizable space. If $X$ is limit point compact, then $X$ is sequentially compact. Proof: Let $(x_n)$ where $n\in \mathbb{N_0}$ be a ...
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1answer
90 views

Prove Heine-Borel Thm

Prove Heine-Borel Theorem: "A subset $S$ of $\mathbb{R}$ is compact if and only if every open cover for $S$ has a finite subcover." Suggestions: Let $S \subset \mathbb{R}$. If every open cover for ...
2
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1answer
79 views

Function for which taking preimages preserves limit points

Suppose we have a surjection $f : X \to Y$ between topological spaces. What is the weakest assumption on $f$, $X$ and $Y$ you can think of that endows $f$ with the following property? If $A ...
0
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1answer
46 views

$B(R,R)$ is not closed in the topology of compact convergence

I'm doing this exercise in Munkres book, and got no clue to solve this problem. Help someone can help me. Let $B(R,R)$ be the set of bounded functions $f: R \rightarrow R$. Prove that ...
0
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2answers
88 views

Show that the metric space is not compact

I have a proof that I would like some hints in solving: Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then ...
1
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2answers
63 views

Compactness of $[0,+\infty)$

Let's say we have $$F = [0,∞).$$ How can we determine whether this is a compact set or not? And let's say we have $U = {(-1,n)}$ ($n∈N$), book said that this $U$ is the open cover of $F$, but I ...
1
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1answer
165 views

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact?

is the vector space $\mathbb{R}^\mathbb{N}$ locally compact? for example, let $x=(x_1,x_2,....)$ any point of $\mathbb{R}^\mathbb{N}$ and let $V=[x_1-\epsilon,x_1+\epsilon] \times ...
3
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1answer
156 views

The product of limit point compact Hausdorff spaces is not limit point compact

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit ...
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3answers
1k views

Proving that the Union of Two Compact Sets is Compact

Prove if $S_1,S_2$ are compact, then their union $S_1\cup S_2$ is compact as well. The attempt at a proof: Since $S_1$ and $S_2$ are compact, every open cover contains a finite subcover. Let the ...
0
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2answers
107 views

Showing a set is not compact by describing an open cover that doesn't have a finite subcover

I would like to prove that the following set is not compact by stating an open cover for it that has no finite subcover. $E=\{x\in\mathbb{Q}:0\leq x\leq2\}$ I'm having trouble thinking of one. A ...
2
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3answers
92 views

Show that the set $[0,2] \setminus \{1\}$ is not compact by exhibiting a cover of open intervals which has no finite subcover.

I think $\bigcup_{k=1}^\infty \left(\frac{-1}{k},2-\frac{1}{k}\right)$ will work, but I'm unsure if the interval includes $2$ as $k \rightarrow \infty$.
3
votes
1answer
120 views

Product of pseudocompact and (sequential) compact is pseudocompact

Let $X$ be a pseudocompact space (i.e. $X$ is Tychonoff space and every continuous function $f :X\to \Bbb R$ is bounded) and let $Y$ be a compact or sequentially compact space. In each case, how to ...
0
votes
3answers
360 views

show that torus is compact

I am having difficuties in showing a torus is compact. Initially I wanted to use Heine-Borel theorem, but after that I realise we are not working in $\mathbb{R}^n$ space. So a simple way to show torus ...
0
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1answer
62 views

compactness theorem and well order [closed]

please, show that well-ordering is not a first-order notion. Suppose that $\Gamma$ axiomatizes the class of well-orderings. Add countably many constants $c_i$ and show that $\Gamma\cup\{c_{i+1} ...
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2answers
521 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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1answer
823 views

Compact spaces and closed sets (finite intersection property)

I am trying to prove the following theorem: A topological space $X$ is compact iff for every collection $\mathscr{C}$, of closed set in $X$ having the Finite Intersection Property (FIP), $\cap C$ of ...
1
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1answer
77 views

Compact spaces and closed maps

If $f: X \rightarrow Y$ is continuous and $X$ is compact and $Y$ is Hausdorff, prove $f$ is a closed map. My attempt: We can consider a subspace $A$ of $X$ and the result is fairly obvious. However ...
1
vote
1answer
65 views

Show that $f$ is onto [duplicate]

Let $(X,d)$ be a compact metric space. Let $f:X\rightarrow X$ be such that $d(f(x),f(y))=d(x,y)$ for all $x,y\in X$. Show that f is onto. Hint: Fix $y\in X $and $ x_1\in X$, define $x_n=f(x_{n-1})$, ...
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0answers
38 views

Equality Constrained Optimization - Conditions to guarantee compactness

Consider a general maximization problem: $X$ and $Y$ compact sets, $f,g: X \times Y \rightarrow \mathbf{R}$, $f,g$ continuous. Problem is $\max_{\substack{x \in X, y\in Y \\ \text{s.t. } ...
2
votes
2answers
99 views

Compactness of an infinite product space.

I'm struggling a little with the idea of compactness. In particular the following question: If we let $\mathbb{N}_n = \{0,1,2,\dots, n\}$ and equip $\mathbb{N}_n$ with the discrete topology then ...
0
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2answers
69 views

Proving sequential compactness from open cover compactness.

Let $(\mathcal M,d)$ be a metric space and $A\subset\mathcal M$. The following types of compactness are equivalent: (i) Each open cover of $A$ contains a finite subcover. (ii) $A$ is sequentially ...
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1answer
149 views

Compact Subsets of $C[a,b]$

Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$ Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm. ...