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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Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact?

Why is $C=\{x\in l^2:|x_n|\leq 2^{-n},n=1,2,3,\dots\}$ compact? I tried to show that $C$ is totally bounded and closed. I showed that is closed but I don't know how to show that is totally bounded. ...
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34 views

A subset of $\mathbb{R}$ is compact iff it is sequentially compact

Let $A \subset \mathbb{R}$. Show that $A$ is sequentially compact if and only if $A$ is compact. I have looked for other explanations of this, but I can't find one that is for $\mathbb{R}$ ...
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47 views

Show that $T^*$ is weak$^*$-to-weak$^*$ continuous

Suppose $Y$ is a separable Banach space and $(y^*_n)_{n \in > \mathbb{N}} \subset B_{Y^*}$. Let $T : X \rightarrow Y$ be an operator. Since $Y$ is separable, we have the unit dual ball ...
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Show that a set is weakly compact nonseparable.

Suppose that $X$ is a nonseparable weakly compactly generated Banach space. Let $W$ be a weakly compact subset which spans a dense linear subspace of $X$. Denote $\mathcal{F}(X) = \overline{span\{ ...
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Why aren't the rationals a compact subset of $\mathbb{R}$?

We define a compact subset of some normed vector space $V$ to be any subset $S$ where every sequence $\{\mathbf{x}_{n}\}$ in $S$ has a subsequence which converges to some $\mathbf{x}$ in $S$. Then ...
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Prove that a continuous image of a closed subset of a compact space is a closed subset

Suppose $f$ is a continuous mapping from a compact metric space $X$ into a metric space $Y$. Prove that if $F$ is a closed subset of $X$, then $f[F]$ is a closed subset of $Y$. Here is my idea ...
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63 views

Are open subsets of compact sets compact?

I'm taking my first course of Analysis and read about this Theorem (2.35) in Rudin: Closed subsets of compact sets are compact. I want to know that whether open subsets of compact sets are ...
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1answer
21 views

Simple Problem on Whether a Set $\Lambda$ is Compact

Using only the definition of a compact set, determine if $\Lambda$ is a compact set. Let $\Lambda = \big[\frac{1}{2},1\big)$ for $n \in \mathbb{N}-\{1,2\}$. Let $\mathscr{F} = ...
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1answer
26 views

Decreasing sequence of closed sets?

Let $(M,d)$ be a compact metric space. Suppose that $(F_n)$ is a decreasing sequence of nonempty closed sets in $M$, and that $\bigcap_{n=1}^\infty F_n$ is contained in some open set $G$. Then $F_n ...
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1answer
56 views

Show the cone on the integers is not locally compact

Let $C\mathbb{Z}$ (the cone on $\mathbb{Z}$) denote the quotient space obtained from $\mathbb{Z} \times [0,1]$ by identifying all points in $\{(z,1)|z\in \mathbb{Z}\}$. How can I show that ...
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Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space

I am trying to prove the following: "Let $U$ be an open subset in the compact metric space $X$. If $\{S_k : k\in\mathbb{N}\}$ is a collection of closed subsets of $X$ such that $S_{k+1}\subset S_k$ ...
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1answer
29 views

Proof compact objects in $\mathrm{Op}(X)$ are compact sets?

The nlab says the compact objects in the category of opens sets of a given space are precisely the compact sets. I'm having trouble with this. I need to show equivalence between $\varinjlim _\alpha ...
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43 views

Disconnectedness, completeness and compactness.

I am in search of examples of metric space which is 1) Complete but not compact 2) Not complete but disconnected 3) Connected but not Complete 4) Compact but not connected. 5) Complete but not ...
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1answer
34 views

Proving intersection (and union) of two compacts sets is compact using subsequence?

Definition: A set $A$ in $\mathbb{R}$ is compact if every sequence in $K$ has a subsequence that converges to a limit that is also in $K$. I want to show that intersection of $A$ and $B$, and the ...
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17 views

Prove closed interval is bounded and thus compact

Prove that the closed interval [1/n, 1-1/n], where n=2,3,4,... is bounded. Isn't it that closed intervals are automatically bounded, and I'm not sure what the intuition is.
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1answer
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Compactness of the unit sphere in finite-dimensional normed vector space

In order to prove that norms defined on any finite-dimensional real (or complex) vector space $E$ are equivalents, I need to proof the compactness of the unit sphere $S_{\infty}=\{x\in ...
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52 views

$f$ is a monotone increasing, but not necessarily continuous, on $\mathbb{R}^n$, $A$ is compact. Is $f$ always has a maximum on $A$?

Call a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$ nondecreasing if $x,y \in \mathbb{R}^n$ with $x \geq y$ implies $f(x) \geq f(y)$. Suppose $f$ is a nondecreasing, but not necessarily ...
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Is there any continuous function $f : \mathbb{R}\to \mathbb{R}$ with $f(\mathbb{R}\setminus K)$ not open for any compact $K$?

I'm trying to find out if there is any continuous function $f : \mathbb{R}\to \mathbb{R}$ such that $f(\mathbb{R}\setminus K)$ is not open for all $K$ compact. Since in $\mathbb{R}$ every compact is ...
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2answers
84 views

If all continuous functions $f: X\subset \mathbb{R}\to \mathbb{R}$ are bounded then $X$ is compact

I'm trying to show that in $\mathbb{R}$ a pseudocompact set is compact. That is, if $X\subset \mathbb{R}$ is such that all continuous functions $f: X\to \mathbb{R}$ are bounded, then $X$ is compact. ...
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1answer
33 views

Show that $K=\{ \frac{x_n}{n} : n \in \mathbb{N} \} \cup \{ 0 \}$ is compact and $X = \overline{span(K)}$

At here, Example $8.2$, there is this statement: Consider any countable and dense subset $\{ x_n : n \in \mathbb{N} \}$ of the unit ball of $X$ and let $K = \{ \frac{x_n}{n} : n \in \mathbb{N} \} ...
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1answer
27 views

Compact set in R^n with maximum metric

I dont know how to solve this: " Show that A is compact, where A is the set of x in R^n which ||x||=1, with ||.|| is the maximum metric in R^n." My try: I know that in R^n: a set is compact iff is ...
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46 views

prove that intersection of the family of the set is connected

I got no clue to solve this problem, because I can't find the connection between the compactness and the connectedness for the set family. Can anyone help me to solve this? I really appreciate. ...
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22 views

The delta values associated with a continuous function

I'll put you in a bit of context. We were proving in class that continuous implies uniformly continuous on compact sets. During the proof an idea came up. When you have a continuous function, you ...
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1answer
23 views

A “Simple Chain of Regions” and Compactness in the Continuum

Let me just start by saying that I'm basically trying to prove this: How to prove every closed interval in R is compact? Except that I need to do it in a very strange way... I'm teaching an ...
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1answer
40 views

Is every countably compact Hausdorff space compact?

It is known that every compact space is countably compact. These properties are equivalent for metrizable spaces. So, is it true that every countably compact Hausdorff space is compact? I think it ...
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53 views

Compactness of $\mathbb{R}^n$

Since $\mathbb{R}^n$ is an infinite space, I thought it is never compact no matter what metric it is endowed with. But, so what is the point in checking for non compactness of $\mathbb{R}^n$ with ...
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1answer
59 views

How to show a set is a norm compact subset

At this paper, page $128$, in Theorem $3.1$, there are two parts which I don't understand how the authors got them: $(1)$ In the first sentence of the proof, why the set $$K=\{ ...
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39 views

Definition of One-point compactification

My question is an elementary one but I can't seem to find the formal definition. If $M$ and $N$ are topological spaces and $\dot{M}$ is the one-point compactification of $M$ then how is a map $f:M ...
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1answer
20 views

If $A$ is clopen in $X$ then $\overline{A}$ is clopen in $\beta X$.

Let $X$ be a Tychonoff space. If $A$ is a clopen subset of $X$, then $\overline{A}$ is clopen in $\beta X$ (of course, $\overline{A}$ means the closure of $A$ in $\beta X$). This is actually the ...
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1answer
43 views

If each $X_n$ is countably compact then so is $\displaystyle\prod_{n\in\mathbb{N}}X_n$.

Let be $X_n$ countably compact and first countable for every $n\in \mathbb{N}$. Then $\displaystyle\prod_{n\in\mathbb{N}}X_n$ is countably compact. I have proved some equivalences. A space ...
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37 views

If $\displaystyle\bigcap_{n\in\mathbb{N}} U_n=\{x\}$, then $\{U_1,U_2,…\}$ is a local base at $x$.

Let $X$ be a countably compact space, $x\in X$, and $U_1,U_2,...$ open sets such that $\displaystyle\bigcap_{n\in\mathbb{N}} U_n=\{x\}$. Then $\{U_1,U_2,...\}$ is a local base at $x$. We take an ...
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1answer
130 views

Can you recommend me a book about compactness (Real Analysis)?

Can you recommend me a book about compactness and connectedness where has examples, because Rudin have a few and are dificult for me. Thanks.
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If $K \subset \mathbb{R}^2$ is compact, then $K$ is closed?

If $K \subset \mathbb{R}^2$ is compact, then $K$ is closed? How I can prove it? Thanks in advance.
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1answer
56 views

$\beta\mathbb{N}$ is embedded in $\beta\mathbb{R}$

I just read that as $\mathbb{N}$ is $C^*$-embedded in $\mathbb{R}$, then $\beta\mathbb{N}$ is embedded in $\beta\mathbb{R}$. A subset $A$ of a space $X$ is said to be $C^*$-embedded if every ...
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80 views

Compactness of a family of closed subsets in a compact space.

Gamelin and Greene's "Introduction to Topology", Second Edition, Dover, pg25, Exercise 8, states that if $X$ is a compact space, then the family of non-empty, closed subsets, $C$, with the metric ...
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1answer
53 views

Continuous bijection is a homeomorphism

Let $X$ be a Hausdorff countably compact space and $Y$ first countably. If $f:X\to Y$ is a continuous bijection then it is a homeomorphism. Like in the case of compact spaces, I'm trying to show ...
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Proving non compactness of a space

I'm trying to show that the space $\mathbb R^p$, endowed with a metric $d'(x,y) = \frac{d_2(x,y)}{1 + d_2(x,y)}$, where $d_2(x,y)$ is the Euclidean distance, is closed and bounded but not compact. ...
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1answer
58 views

Prove that a decreasing sequence of compact sets converges in the Hausdorff metric

Let $A_n$ be a decreasing sequence of nonempty compact sets in $X$. Then I want to prove that: $$\displaystyle \lim A_n = \bigcap_{n\ge 1} A_n$$ in $\mathcal C_H$. Where $$C_X=\{\text{non empty ...
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1answer
44 views

star compact Hausdorff is countably compact

I have read in some papers stated that every star compact Hausdorff is countably compact. I don't know how to prove it. Note: A space $X$ is called star compact if for any open cover $\mathcal{U}$ of ...
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1answer
30 views

Set of all contraction maps

Consider $(X, d)$ be a compact metric space, and let $Con(X)$ denote the set of all contraction maps on $X$. We shall define the “distance” between two maps $f, g ∈ Con(X)$ as follows, $$d_{Con(X)}(f, ...
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A line is simply connected but a circle is not. Does this mean that the one-point compactification of the real line is not simply connected?

I was reading about end compactification, following up my interest in compactification of the real line, Homeomorphism of a compact real line to the real line. Question 1&2: Is a Warsaw circle ...
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Image of a precompact under the action of uniformly continuous function is a precompact

Suppose we have two metric spaces $(X, \rho_x)$ and $(Y, \rho_y)$ and a uniformly continuous function $f\colon X \to Y$. The problem is to prove that image $f(A)$ of every precompact $A \subset X$ ...
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1answer
31 views

What is the definition of sequential precompactness?

I know that topological space $X$ is called precompact if any sequence in $X$ has a subsequence convergent in X. In my book of calculus of variation I have encountered the word sequential ...
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Boundedness in $W^{-1, \infty}_{loc}$

For the solution of vanishing viscosity equation: $u^{\epsilon}_{t} + f(u^{\epsilon})_{x} = \epsilon u^{\epsilon}_{xx}$ in $\mathbb R$ X $R^{+}$ with bounded measurable initial data: ...
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1answer
62 views

Without using the Heine-Borel Theorem, show that a closed subset A of a compact set K is compact.

So how can I start this without using this theorem? I know that A needs to be closed and bounded.
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29 views

Do we need the surjectivity of a continuous $f: (X, \tau) \to (Y, \rho)$ to get $(X, \tau)$ compact, implies $(Y, \rho)$ compact?

I found the following proposition in Morris' "Topology without tears": Proposition: Let $f: (X, \tau) \to (Y, \rho)$ be a continuous surjective map. Thus, if $(X, \tau)$ is compact, then $(Y, ...
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1answer
33 views

closed function with image compact hausdorff is continuous

I need to prove that if $f:X\rightarrow{Y}$ is closed and $Y$ is compact Hausdorff, then $f$ is continuous. I tried proving it with the pre image of a closed set, and proving it is closed, but I got ...
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1answer
64 views

Proving the compactness of a subset of a compact set

I have to prove the fact that, given a metric space $(X,d)$ and a subset $K$ of $X$ compact, taking a closed subset $C$ of $K$, this $C$ is compact too. I have used the characterization of closed ...
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1answer
40 views

Continuous map on $\ell^p$ and Compact subset

Let $ 1 \leq p < \infty $. Consider the normed space $\ell^p$ . Show that the following map is continuous. $$T(x_1,x_2,\ldots,x_n,\ldots) = (x_1^2,x_2^2,\ldots,x_n^2,\ldots) $$ Now let $p=1$. ...
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1answer
37 views

Isometric problem using compact spaces

Let $A\subset \mathbb{R}^{n}$ a compact subset and $f:A\rightarrow A$ an isometry (i.e , a function such that $\| f(x)-f(y)\| = \| x-y\|$ for all $ x,y \in A$). Show that $f(A)=A$ Now, I've already ...