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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1answer
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Is there ever an example of when the union of a space $X$ and a collection of compact subspaces is not compact?

I have seen several examples where the union of a space $X$ and collection of compact subspaces is compact. Is there ever an example of when the union of a space $X$ and a collection of compact ...
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1answer
21 views

Is $\{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ compact in $(\mathbb R^2, \mathcal E_2)$?

Determine if $X = \{(x,y) \in \mathbb R^2 \mid x^2 + x^3y^2 = 0\}$ is compact in $(\mathbb R^2, \mathcal E_2)$, where $\mathcal E_2$ denotes the standard Euclidean topology. I know that $X$ is ...
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1answer
42 views

Does being Nonempty Compact Set on $\mathbb{R^+_2}$ imply being Convex set?

Look at the domain of a function $y=x-2$ where $x\in\mathbb{R_+}$. Then, the triangle produced by x and y-intercepts is bounded and closed. So it is compact. Suppose it is also nonempty. Does this ...
3
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1answer
42 views

Intersection of Compact sets Contained in Open Set

Just wanted to see if my proof of the following is valid: Let $\{K_i\}_{i=1}^{\infty}$ be compact sets (in some metric space), and let $V$ be an open set such that $$ \bigcap_{i=1}^{\infty} K_i \...
2
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1answer
396 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
2
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1answer
24 views

Complement of compact subspace of surface

Let $X$ be a smooth 2-manifold, $K$ be a compact subset of $X$, such that only one component of $X\backslash K$ does not have compact closure, call this component $U$ (there may be other components). ...
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3answers
37 views

Compact set and a sequence of closed sets, the intersection of all of them is empty

I'm having a hard time to prove this. The problem is: Let $V \subset \mathbb{R}^d $, be a nonempty compact set and $(A_n)_{n\in\mathbb{N}}$ a sequence of closed nonempty sets in $\mathbb{R}^d$ with ...
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2answers
64 views

$M\times N$ compact $\implies$ $M$ compact and $N$ compact

I must prove that $M\times N$ compact $\implies$ $M$ compact and $N$ compact using the definition that, if a metric space $M$ is compact, then every cover has an open finite sub cover. $$M=\cup ...
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2answers
45 views

Compactness of the set of points where a continuous function achieves a local maximum

Let $(K,d)$ be a compact metric space, and $f:K\rightarrow \mathbb{R}$ be a continuous function on $K$. Define: $$M=\left \{ x\in K :\text{$f$ achieves a local maximum in $x$} \right \}$$ I need to ...
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0answers
31 views

Generalization of Strict Local Maxima

I try to generalize a strict local maximum to a local roof which can possibly be a flat area instead of just a single point. Below is my attempt: Let $f$ be a continuous real-valued function on $R^D$ ...
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1answer
22 views

Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
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0answers
29 views

Uniform convergence of a equicontinuous sequence of functions [duplicate]

Let $X$ be a compact metric space, and $(C(X),d_{\infty})$ the space of continuous functions. Let $D\subset{X}$ be a dense subset, and $\{{f_n\}}_{n \in N}$ a equicontinuous sequence of functions from ...
3
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0answers
36 views

Comparing different definitions of tightness for measures

Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure. Consider the following properties: (1) $\forall A \in \mathcal{B}(X): \...
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1answer
61 views

Continuous map on $S^2$

Can you help me with this? Let $S^2 := \{x\in \mathbb R^3:||x||_2 = 1\} \subset (\mathbb R^3, ||\cdot||_2)$ and $T:S^2 \to (\mathbb R, |\cdot|)$ be a continuous map. a) Why does T assume its ...
3
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1answer
57 views

Question about the proof that the Hilbert Cube is compact.

Because of the fact that $(1)$ The topological space $[0,1]$ is a continuous image of the Cantor space $(G,T)$. There exists a mapping $\phi_n$ of $(G_n, T_n)$ onto $(I_n, T'_n)$ where, for each ...
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1answer
32 views

Checking the compactness of sets

I have to check to following sets for compactness in the given spaces with respect to the standard norm for them: \begin{align} M_1 &:= \{f_n:\left[-1, 1\right]\rightarrow \Bbb{R}| f_n(x) = n \cos(...
3
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1answer
63 views

core-compact but not locally compact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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2answers
63 views

Show that a subset $E$ $\subset \Bbb Q$ is not compact in $(\Bbb Q, d)$ and decide whether it is open or not

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} = \{p \in \Bbb Q : \sqrt2 < p < \sqrt3\} \subset \Bbb Q.$ I have to show ...
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5answers
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Projection map being a closed map

Let $\pi: X \times Y \to X$ be a projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is ...
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1answer
52 views

Prove there is a compact self adjoint $S:H\to H$ such that $S^3=T$.

Let $T:H\to H$ be compact and self adjoint. Prove there is a compact self adjoint $S:H\to H$ such that $S^3=T$. Is the $S^3$ means power of 3 or applying the operator 3 times? What is there to prove ...
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1answer
42 views

$\{M \in \mathcal{P}(\Bbb Z): \:\: M=\emptyset \: \textrm{or } (-13\in M \wedge13 \not \in M )\}$ give characterisation of compact, non-finite sets

Let $\mathcal T_{\Bbb Z}$ be the following topology on $\Bbb Z$: $$\mathcal T_{\Bbb Z}:=\{M \in \mathcal{P}(\Bbb Z): \:\:\: M=\emptyset\: \: \textrm{or } M=\Bbb Z \: \: \textrm{or } (-13\in M \...
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1answer
40 views

On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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2answers
77 views

$f$ be a function on real line carrying compact sets to compact sets and fiber of every point under $f$ is closed , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R$ be a function such that it carries compact sets to compact sets and $f^{-1}(\{x\})$ is closed for every $x \in \mathbb R$ , then is $f$ continuous ? (I know that if $...
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2answers
48 views

Uniform convergence of n-fold composition using Schwarz lemma

Let $f$ be an analytic function mapping the unit disk $\mathbb D$ to itself with $f(0) = 0$ and $|f'(0)| < 1$. Let $f^{n} = f \circ f \circ \dots \circ f$ be the function obtained by composing $f$ ...
0
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1answer
36 views

In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
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1answer
61 views

Is this set $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$ compact?

Is this set $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$ compact? I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph of ...
3
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2answers
166 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
3
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1answer
44 views

Prove that an infinite chain of proper containments of compact sets is non empty [duplicate]

I need to prove that if $K_1\supset K_2 \supset K_3 \supset K_4 \supset \ldots$ is a chain of proper containments and each $K_{i}\subseteq \mathbb{R}^{n}$ is compact, then $\bigcap_{i=1}^{\infty} K_{i}...
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2answers
56 views

Is a complement finite topological space compactly generated?

I'm using the following definition: A topological space $X$ is called compactly generated if it verifies: Any subset $U$ of $X$ is open iff for any Hausdorff compact space $Y$ and continuous map $f:Y\...
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4answers
91 views

The two definitions of a compact set

In general, $A$ is compact if every open cover of $A$ contains a finite subcover of $A$. In $R$, $A$ is compact if it is closed and bounded. The second is very easy to understand because I can ...
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3answers
56 views

$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
2
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1answer
35 views

Tychonoff's theorem for completely regular spaces and the axiom of choice

It is well-known that Tychonoff's theorem, i.e., that the product of any set-indexed family of compact spaces is compact, is equivalent to the axiom of choice. It is also the case that if the spaces ...
0
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1answer
23 views

Does there exist any surjection from a Hausdorff compact space to complement finite topological space?

Let $X$ denote a countable infinite set endowed with complement topology (that is, $X$ itself and all its finite subsets consisit of the set of closed set). Can we find a Hausdorff compact space $Y$ ...
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1answer
34 views

$\ell_0$ norm and the induced complete metric spaces

I have been reading about the $\ell_0$ norm, wikipedia gives us that "The mathematical definition of the $\ell_0$ norm was established by Banach's Theory of Linear Operations. The space of sequences ...
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0answers
42 views

A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
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2answers
32 views

Finite product of compact subspaces is compact

I am working through an old qualifying exam and I have run into what I think is a very simple question but the way it is posed is confusing to me and I am worried I a missing the point. The question ...
0
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1answer
49 views

Homeomorphism between a compact space and non compact space? [closed]

Can such a thing exist? I am relatively new to topological spaces but my intuition would be that such a thing cannot exist. I guess this question could also be asking whether a Hausdorff space can be ...
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0answers
23 views

Closed unit ball in $\ell^p$ is compact

I'm curious as to whether the closed unit ball in $\ell^p$ is compact for $1 \leq p \leq \infty$, with respect to the $p$-norm, and the $\sup$ norm?
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1answer
36 views

Sequences of 0s and 1s are compact

Let $X$ be the space of sequences $x = (x_1, x_2, ..., x_n,...)$ such that $x_i = 0$ or $x_i=1$, equipped with the metric $$d(x,y) = \sum_{n=1}^{\infty} \frac{1}{2^n} \left| x_n - y_n \right|.$$ Prove ...
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0answers
53 views

Is there a name for a topological space $X$ in which every proper closed subset is compact? [duplicate]

Is there a name for a topological space $X$ in which every proper closed subset is compact$^{(*)}$? It is well known that in a compact topological space, every closed set is compact. Hence, the class ...
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1answer
34 views

Is any closed ball compact in the Weak$^*$ topology $\sigma(E^*,E)$ for a Banach Space $E$?

For a Banach Space $E$, the Banach Alaoglu Bourbaki theorem asserts that the closed unit ball in $E^*$: $$B_{E^*}= \{f \in E^* \ | \ ||f|| \leq 1 \} $$ is compact in the weak$^*$ topology $\sigma(E^*...
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1answer
52 views

Showing $f$ has a maximum [closed]

Assume that $(X,d)$ is a metric space and that $f: X \to [0, \infty)$ is a continuous function. Assume that for each $\epsilon >0$ there is a compact set $K_{\epsilon} \subset X$ such that $f(x) &...
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2answers
67 views

Showing the continuity of $d(x,f(x))$

Assume that $(X,d)$ is compact, and that $f: X \to X$ is continuous. Show that the function $g(x) = d(x,f(x))$ is continuous and has a minimum point. Consider the function $g(x) = d(x,f(x))$. If $g$ ...
4
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1answer
127 views

Compactness and sequential compactness in metric spaces

I've got a question: I'm trying to prove that every metric space is compact if and only if the space is sequentially compact. In all the proofs I have found, they used the Bolzano-Weierstrass theorem. ...
2
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0answers
35 views

Compactness and sequential compactness are equivalent in metric space but not always in others

There are many discussions about such question; however, they are proof-type answer Compactness and sequential compactness in metric spaces If $(X,d)$ is a metric space then I want to show ...
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2answers
37 views

Which of the following are compact I need Hint…

Which of the following are compact? $\{(x,y) \in \mathbb{R}^2 :(x-1)^2+(y-2)^2=9\} \cup \{(x,y) \in \mathbb{R}^2: y=3\}$. 2.$\{(\frac{1}{m},\frac{1}{n}) \in \mathbb{R}^2:m,n\in \mathbb{Z}-\{0\}\} \...
2
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1answer
29 views

A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection

I came across the following proposition in my Complex Analysis book, stated as "A set $K \subseteq X$ is compact iff every collection $\mathcal{F}$ of closed subsets of $K$ with the f.i.p. has $\cap_{...
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1answer
17 views

Motivation and intuition behind concept of totally bounded-ness in metric spaces

The way I understand it is, a set in a m.s. is totally bounded means the set Admits a finite open cover of fixed size. Regardless of whether it admits an arbitrary open cover. Why do we need totally ...
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2answers
28 views

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [closed]

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism Requirements for a homeomorphism $f:X \rightarrow Y$: $f$ is ...
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0answers
24 views

Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...