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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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 ...
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42 views

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 ...
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Tips on determining compact to non-compact sets

I find it hard to quickly determine this. For instance, $D^n/\{0\}=\{x \in \mathbb{R}^n| 0 < ||x|| \leq 1\}$ is non-compact. $S^{n-1}=\{x \in \mathbb{R}^n | ||x||=1\}$ is compact. So, ...
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26 views

calculating the diameter of a set

I am trying to prove that if $X$ is complete and $ M \subset X$ has a finite epsilon net, then $M$ is relatively compact. I have the proof to hand: I don't understand how they calculated that ...
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124 views

Prove expansive function on a compact set is surjective.

Let $M$ be a compact set and $(M,d)$ be a metric space, define function $f:M\to M$ such that for all $\,p,q\in M$ $$d(f(p),f(q))\ge d(p,q)$$ Prove $f$ is surjective. I observed that compactness ...
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Given $A \subset \mathbb{R}$ is bounded, what is an open cover of $A$?

I am looking for a good example of a covering of a bounded set $A$ in $\mathbb{R}$ Currently my example is $\{I_n\} = \{(k + n - \frac{\epsilon}{2^n}, k + n - \frac{\epsilon}{2^n})\}_{n \geq 0}$, ...
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23 views

Orientation of a triangulated compact surface, using orientations of triangles

The questions I am working on asks me to :"Give the definition of an orientation of a triangulated compact surface by using orientations of triangles" I know that a surface is orientable if the ...
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Is this basic function space compact?

Let $A=L^2(X)$ be the space of square integrable functions on a compact Euclidean space $X$. If we equip $A$ with the usual 2-norm, is $A$ compact? Edit: And if we restrict AA by adding the ...
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Intersection of nested compact sets in a Hausdorff space

Suppose that a non-empty Hausdorff space $X$ is compact and there is a continuous map $f:X \to X$. Let $X_1=X$ and put $X_{n+1} = f(X_n)$ inductively for all $n \in \mathbb{N}$. Prove that $A ...
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41 views

Show that the special unitary group $SU(n)$ is a compact topological group

What I know: $SU(n)=${$A \in U(n): detA=1$} where $U(n)=${$n \times n$ matrices $A: AA^*=I=A^*A$} with elements in $\mathbb{C}$ and $A^*$ is the complex transpose of $A$ A topological group is a ...
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55 views

Does it make sense to define a “metric topological space” $(M, d, \tau)$

When doing things related to compactness, sometimes you have to switch definition from sequential compactness which is defined on a metric space $(M, d)$, to things related to covering compactness ...
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limsup definition in Alexandroff compactification of C

In the Alexandroff, or one-point compactification of $\mathbb{C}$ (by adding a point $\infty$), consider a function $f:\Omega (\subset \mathbb{C}) \to \mathbb{R}$. I was asked to justify the following ...
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1answer
62 views

Proof of the Arzelà–Ascoli Theorem

I'm stuck on a particular line of the proof of The Arzelà–Ascoli Theorem. In lectures, we have: $1.$ Defined equicontinuous as: Let $X$ be a metric space, $C(X) = \{f: X \rightarrow ...
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1answer
46 views

Given $f:\mathbb R\to[0,1]$ and a nonnegative sequence $\{u_n\}$, show that $\{f(u_n)\}$ has a convergent subsequence.

Let $f$ be a function from $\mathbb R$ to $[0,1]$. Prove that if $(u_n)$ is a sequence of non-negative real numbers, then there exists a subsequence $(u_{n_k})_{k\in\mathbb N}$ such that ...
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31 views

$A, B \subset X$ compact, disjoint subsets of a Hausdorff space. Show existence of disjoint empty sets

Problem: Let $A, B \subset X$ be both compact, $X$ is a Hausdorff space, with $A \cap B = \emptyset $. Show that there exists open sets $U, V \subset X$ with $A \subset U, B \subset V$ and $U \cap V ...
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Prob. 17, Chap. 2 in Baby Rudin: The set of all numbers in $[0,1]$ with only $4$ and $7$ as decimal digits is countable, dense, compact, perfect?

Here is Prob. 17 in the Exercises after Chapter 2 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $E$ be the set of all $x \in [0,1]$ whose decimal expansion ...
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Is $\beta\omega$ hereditarily irresolvable?

$X$ is called resolvable if it can be represented as a union of two disjoint dense sets, it is irresolvable otherwise. Moreover it is hereditarily irresolvable (HI) if every subspace of X is ...
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1answer
21 views

Show that every paracompact space is nearly paracompact but the converse is not true

I am learning about the para-compact space and nearly para-compact space. I know that every nearly para-compact space is para-compact space but the converse is not true in general. So i need an ...
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1answer
49 views

Prob. 13, Chap. 2 in Baby Rudin: Construct a compact set of real numbers whose limit points form a countable set

Here's Prob. 13 in the Exercises after Chap. 2 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Construct a compact set of real numbers whose limit points form a ...
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51 views

Z compact with another topology?

Consider $\mathbb{Z}$ equipped with a topology generated by the basis of the form $(-k,k)$, is a finite set, say (-3,3), compact? Is $\mathbb{Z}$ compact? I think a finite set is compact since you ...
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1answer
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Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$

Let $\{K_i\}_{i=1}^{\infty}$ a decreasing sequence of compact and non-empty sets on $\mathbb{R}^n.$ Then $\cap_{i = 1}^{\infty} K_i \neq \emptyset.$ I heard about a proof that take $x_i \in K_i.$ ...
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Why isn't every subset of a compact set also compact?

I've seen on MathSE that a subset of a compact set need not be compact. However, I am not fully understanding why. If $X$ is compact then there exists a finite subcover, i.e., $X \subseteq ...
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1answer
24 views

Separating the supports of disjoint continuous functions

Can the supports of disjoint continuous functions on a compact Haussdorf space always be separated by open sets? I.e.: given a compact Haussdorf space $X$ and a sequence of continuous functions ...
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25 views

Compactness in Hilbert spaces

Let $H$ be a Hilbert space with orthonormal basis $\{h_n:n\in \Bbb N\}$. Let $P_n$ be the orthogonal projection to $\operatorname{span}\{h_1,\cdots, h_n\}$. Claim: A bounded subset $U\subset H$ is ...
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38 views

show a set is open

Suppose X and Y are compact and they are Hausdorff topological spaces. Let $f : X \rightarrow Y$ be a continuous surjective function. Prove that any $U \subset Y$ is open if and only if $f^{−1}(U)$ is ...
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Cartesian Product of Compact Set and Non-Compact Set is Non-Compact

Theorem: Let $A$ be a compact set and $B$ be a non-compact set. Then $A\times B$ is non-compact. I know that if $B$ is non-compact, then there exists an open cover $O$ of $B$ that does not have a ...
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1answer
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Suppose $X$ is normal space and $\mu$ is a Radon measure, then the union of open null sets is again a null set. [closed]

I tried to prove this statement by contradiction and basic topology tricks, but failed.
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Let $A\subset \Bbb R^2 $ with the property that every continuous function on $A$ has a maximum in $A$ .Prove that $A$ is compact.

Let $A\subset \Bbb R^2 $ with the property that every continuous function on $A$ has a maximum in $A$ .Prove that $A$ is compact. My try: We have to show that $A$ is closed and bounded. In order to ...
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Failing to recognize the importance of compactness, connectedness, and other topological notions in Real Analysis [closed]

I'm currently taking a course in Real Analysis that uses Principles of Mathematical Analysis by Rudin, and having a somewhat difficult time on tests. I always read that notions like compactness, ...
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Is the set of functions of the form $\int_a^x f(t) dt$, where $f(t)\in A$, which is a bounded subset of $C_{[a,b]}$, closed?

Let $A$ be a bounded subset of $C_{[a,b]}$. I want to show that the set of functions of the form $\int_a^x f(t) dt$ for $f(t)\in A$ is compact (this problem appears in Kolmogorov & Fomin's ...
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79 views

Compact subsets of metric space with French railway metric

Let $A=\{0,1,2,...\}$ with $f$ the French railway metric that has centre $0$ and $f(a,0)=1$ for all $a\in A$ with $a\neq0$. How do I show that the metric space $(A,d)$ is complete? How do I ...
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1answer
53 views

Show that every nearly compact space is almost compact space but the converse is not true

I am learning about the almost compact space and nearly compact space. I know that every nearly compact space is almost compact space but the converse is not true in general. So i need an example of ...
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70 views

Motivation of paracompactness

"A paracompact space is a topological space in which every open cover admits a locally finite open refinement" is the definition of paracompactness on Wikipedia. Comparing with the definition of ...
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1answer
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Second countable spaces under continuous closed surjective maps

Let $p:X\rightarrow Y$ be a closed continuous surjective map such that $p^{-1}(y)$ is compact for each $y\in Y$. Show that if $X$ is second countable then $Y$ is second countable. Let $y\in Y$ and ...
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A sequentially compact metric space is bounded. Help me fix this proof.

I know this is usually done by contradiction but I'm trying out something a bit different: Let $\mathbb{X}$ be a sequentially compact metric space. Let $(s_n)$ be a sequence in the metric space such ...
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33 views

The distance between disjoint closed sets may be zero [duplicate]

Let $K$ and $L$ be nonempty compact sets, and define $$d= \inf \{|x-y|: x \in K \wedge y \in L\}.$$ Show that it is possible to have $d = 0$ if we assume only that the disjoint sets $K$ and $L$ ...
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1answer
26 views

Counterexample of intersection characterization of compactness

Given metric space ($X,d$), let {$S_1, S_2,...$} be a set of non-empty sets where $S_1 \supseteq S_2...$, then if $X$ is compact and the $S_t$ are closed then $ \cap_tS_t$ is not empty. In $\Bbb R$, ...
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Is the fact true that compact set is in perfect set?

By definition, Perfect set $E_1$ is closed set without isolated points. Compact set $E_2$ is bounded and closed set in Euclidean space; $\mathbb{R}^n$. Is the following equation true? $$E_2 ...
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Definition request: explicit definition of covering compactness in terms of set notation

Part of my confusion with covering compactness stems from the fact that it is a definition given almost completely in a high level manner (in English no less). When I look at: A set $A \subset ...
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67 views

Show that $\int_{a}^{b}{x^{n}f(x)dx}=0$, then $f=0$

Let $f:([a,b],\vert\vert)\to (\mathbb{R},\vert\vert)$ a continuous function. Show that, if $$\int_{a}^{b}{f(x)x^{n}dx}=0$$ for all $n\in\mathbb{N},n\geq 0$, then f is identically zero. My ...
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51 views

What does “closed” mean in Heine Borel for $C^0$?

Heine Borel for $C^0$: A set $\mathcal{E} \subseteq C^0([a,b], \mathbb{R})$ is compact if it is closed, bounded and equicontinuous. I don't really understand what closed mean in the ...
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2answers
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For a compact metric space $X ,f: X \rightarrow X $ s.t $ d(x,y)\leq d(f(x),f(y))$ is surjective [duplicate]

I want to show that in a compact metrix space $X$ ,the function $f :X \to X$ such that $d(x,y) \le d(f(x),f(y))$ is surjective! I tried to show that f is continuous and injective but i don't think it ...
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What is wrong with this argument that closed interval [0, 1] is not compact?

I am a student majoring engineering. I am studying real analysis with textbook 'Measure and Integral' by Wheeden and Zygmund. This book defined compact like the following: $E$ is compact if ...
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2answers
54 views

An application of Urysohn's lemma

Let $X$ be a Compact Hausdorff space. Assume that the vector space of real valued continuous functions on $X$ is finite dimensional. Show that $X$ is finite. Suppose $X$ is infinite, given any $n\in ...
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1answer
50 views

Proving that continuous image of a closed bounded subset of $\mathbb{R}$ is closed and bounded (without compactness)

I wish to prove that continuous image of a closed bounded subset of $\mathbb{R}$ is closed and bounded (the function is $\mathbb{R}\rightarrow\mathbb{R} $). However, I do not have the equivalence to ...
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1answer
24 views

Graph of function is compact

Let $X$ be a Hausdorff space. Let $f:X\rightarrow \mathbb{R}$ be such that $\{(x,f(x)):x\in X\}$ is a compact subset of $X\times\mathbb{R}$. Show that $f$ is continuous. What i have done so far is : ...
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1answer
24 views

Which statement is true for non-compact sets in a metric space [closed]

We know that a set is compact if for every open cover, there exists a finite subcover. If a set is not compact then is it true that: There exists an open cover, such that there does not exists a ...
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1answer
25 views

Compact Metric Spaces & Triangle Inequality Theorem [duplicate]

Let X be a metric space, p ∈ X, and let K ⊂ X be compact. Show that there exist x0, x1 ∈ K such that d(x0, p) ≤ d(x, p), ∀ x ∈ K, d(x1, p) ≥ d(x, p), ∀ x ∈ K. I know that I have to show the distance ...