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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1answer
59 views

One point compactification of $[0,1]\times [0,1)$

What is one point compactification of $[0,1]\times [0,1)$? If we draw the figure we see that top line is missing and we've to add just one point to make it compact. So I think triangle will be ...
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1answer
46 views

Is $C[0,1]$ locally Compact?

I'm asked to use the function $f_n(x)=nx$ for $0\le x\le \frac{1}{n}$ and $f_n(x)=1$ for $\frac{1}{n}\le x\le 1$. I'm not familiar with Functional Analysis.
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0answers
38 views

Question about compact sets in $\mathbb{R}$

Suppose I am given a function $f$ on $\mathbb{R}$ and I'm asked to show that $ \int_K f(x) \,dx < \infty$ for any compact set $K \subset \mathbb{R}$. Would it be enough to only consider the ...
0
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1answer
52 views

Compactness of second-countability of $\omega$X$\omega_1$

Please discuss the following properties of the product space consisting of $\omega$X$\omega_1$: Is it compact? Is it 2nd countable? $\omega$ is the first infinite ordinal and $\omega_1$ is the ...
0
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1answer
38 views

A continuous integer-valued function on a compact metric space has finite range

Let $X$ be a compact metric space and let $f:X\to\mathbb Z$ be a continuous function. (Here $\mathbb Z$ has the Euclidean topology induced from $\mathbb R$.) Prove that $f$ can assume only finitely ...
2
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0answers
35 views

Properties of Hilbert Spaces- Contrasting Two Different Topological Spaces

Let H be the space of real sequences x = $(x_1 , x_2, ... )$ with $\sum(x_n^2)$ finite. (This is $l_2$ in fact.) I wish to show the following: The topology on H is ...
0
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1answer
124 views

Are the following topological spaces locally compact?

I am trying to determine whether the following spaces are locally compact: a) the slotted plane b) the radial plane For part a) I am almost certain that it is not compact, but not sure how to go ...
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1answer
34 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
3
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3answers
93 views

Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.5, Problem 10

Here's Problem 10 in Section 2.5 in Introductory Functional Analysis With Applications by Erwin Kreyszig: Let $X$ and $Y$ be metric spaces, let $X$ be (sequentially) compact, and let the mapping ...
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0answers
41 views

Induced topology by a complete uniform space.

I know that Uniform space is generalization idea of metric space,Uniform space like metric space induce a topological space. Now my question is ( or are ):- In case our Uniform space was complete ...
0
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0answers
30 views

Question related to Uniform Space

I have questions related to Uniform Space; If $X$ is a countable discrete space, then how to show that finest pre compact uniformity on $X$ admits a countable base of entourages. If $\mho$ is a ...
0
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1answer
51 views

Cantor Intersection Theorem Without Closedness, counterexample

The Cantor Intersection Theorem is that Let $\{S_1,S_2,S_3,...\}$ be a countable collection of nonempty sets in $\mathbb R$ such that: $S_{k+1} \subset S_k$ for $k=1,2,3...$ Each $S_k$ ...
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2answers
57 views

Characterization of compact subsets in the metric space of all complex-valued sequences

Here's the statement of the Problem 4 after Section 2.5 in Introductory Functional Analysis With Applications by Erwine Kryszeg: Show that for an infinite subset $M$ in the space $s$ to be ...
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1answer
40 views

Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
3
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2answers
140 views

Decreasing sequence of compact subsets of a Hausdorff space

Let $E$ be a Hausdorff topological space and $(K_{n})_{n \in \mathbb{N}}$ be a decreasing sequence of compact subsets of $E$. Let $U \subset E$, $U$ open with $\bigcap_{n \in \mathbb{N}} K_{n} \subset ...
1
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1answer
63 views

Proving a subspace of $l^1_\infty$ is compact

Any help on this would be appreciated. I'm trying to prove that the subspace $(E,\rho)$ is compact. $$E = \{\{x_n\}_n \in X: |x_n|\leq1/(3^n)\text{ for every }n\}$$ $$X=\{\{x_n\}_n \in X: \sum ...
0
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1answer
33 views

AC and Tychonoff theorem

Although I have proof with me that Tynhonoff theorem implies AC. But I have some difficulties with it: 1. Do we define topology on empty set. If not then in proof of Tynhonoff theorem implies AC we ...
1
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1answer
25 views

Subspace of Paracompact space.

Are subspace of a paracompact space is normal? This is what I think about this question... First A paracompact space+ Hausdorff turn out to be Normal, second the paracompact property is not ...
3
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0answers
22 views

Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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1answer
73 views

Proving uniform convergence of an integral-defined function on compact sets

If $f$ is a compactly supported smooth (infinitely differentiable) function into $[0, 1]$ such that $\int f(x)dx = 1$, $g$ is a continuous function, and $f_\epsilon(x) = ...
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1answer
58 views

Clarification: Every infinite subset of E has a limit point of E iff E is compact.

Every infinite subset of E has a limit point of E iff E is compact. Is this always true?
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0answers
78 views

Lindelöf Property and Compact space

Let $X$ be a compact space and $L$ is the smallest family of subspaces of$\,X\,$that contains all closed sets and is closed with respect to countable union and intersection. The question is :- Is ...
0
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1answer
52 views

Compactness and cartesian product

I'm having trouble figuring out how can I show that if two sets are compact then their cartesian product is also compact. Any help is much appreciated,thank you!
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0answers
37 views

A problem of a Hausdorff space $X$ that is locally compact at the point $x$

Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each open neighborhood $U$ of $x$, there is an open neighborhood $V$ of $x$ such that $\operatorname {cl}(V)$ ...
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2answers
48 views

An example of a not locally compact space in $\mathbb R^2$

Are the two subspaces $X$ and $\operatorname{cl}(X)$ of Euclidean space $\mathbb R^2$ locally compact? $$X = \{(x,\sin 1/x) \mid 0 < x \le 4\}\cup\{(x,\sin 1/x) \mid -4 \le x \lt 0\} \cup ...
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1answer
48 views

A problem in locally compact Hausdorff space

I am trying to solve the following problem. Let $X$ be locally compact Hausdorff and $Y$ be Hausdorff. (a) If $f: X \to Y$ is continuous and open map then show that $f(X)$ is locally compact. (b) ...
2
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1answer
58 views

Which of the following are compact?

Which of the following are compact? $1$. The set of all upper triangular matrices all of whose eigenvalues satisfy $|\lambda|\leq 2$. $2$. The set of all real symmetric matrices all of whose ...
2
votes
1answer
22 views

Example of continuous functions $f\colon S \to T$ such that $f(S)=T$.

I would like to find an example of a continuous function from $S=(0,1)$ to $T=(0,1)\cup (1,2)$ such that $f(S)=(T)$. At the moment the only thing I can think might work would be to check whether ...
3
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0answers
103 views

A question about compact sets: how to prove $g$ must be an isometry [duplicate]

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
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3answers
58 views

Let (X, d) be a metric space and A, B ⊂ X be two compact subsets. Show that A ∩ B is also compact

Question seems fine i just have a few doubts. Is it possible to just use the Heine Borel theorem? as both A and B are compact it implies they are both closed, so therefore their intersection is ...
0
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1answer
61 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
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1answer
32 views

Show compactness of a set given by inequalities

Show that the subset $A=\{(x_1,...,x_n)\in\Bbb R^n |−1≤x_1 ≤x_2 ≤···≤x_n ≤1\}$ is compact. A is contain in an open cover as it is contained in $\Bbb R^n$. Therefore there exists a finite sub cover ...
3
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2answers
51 views

Characterising the discrete topology with compact subsets [duplicate]

If a set is endowed with the discrete topology then a subset is compact iff it is finite. Is the converse true? That is, given a Hausdorff topological space such that every compact subset is finite, ...
0
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1answer
46 views

Easy question about compactly contained sets

Let $U\subseteq X$ be a an open set of a topological space $X$ and $V\subset\subset U$ an open, compactly contained set (i.e, $\bar{V}$ is compact and $\bar{V}\subset U$). When we say the closure of ...
3
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2answers
35 views

Does one corona set project onto the other?

Let $X$ be a locally compact Hausdorff space. By a compactification of $X$, let us understand a pair $(C,\iota)$ consisting of a compact Hausdorff space $C$ and a topological embedding $\iota : X ...
0
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1answer
43 views

Is every continuous one-to-one image of $[0,\infty)$ locally compact?

Suppose $f:[0,\infty)\to Y$ is continuous and one-to-one onto $Y$. You may assume $Y$ is metric. Is $Y$ locally compact? Thanks!
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0answers
70 views

Cantor Space - Example - Proving Compactness/Perfectness/Closed/Totally disconnected

Say we take the set of infinite binary codes $\{0,1\}^\mathbb{N}$, which is often written as $2^\mathbb{N}$, mapped to the Cantor set defined previously as $C_n=\frac{c_{n-1}}{3} \cup \left( ...
3
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1answer
109 views

Zero sets in Mrówka spaces

For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in ...
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2answers
174 views

A set A is compact, is its boundary compact?

I am trying to understand the concept of a boundary, and I have seen it defined $Bd(A) = \overline{A} \cap \overline{A^{\complement}}$. I was wondering three things, First how can I show that the ...
2
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1answer
122 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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2answers
42 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
0
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1answer
66 views

if $X$ has a finite number of isolated points, is $X$ compact?

If every real valued continuous function on $X$ is uniformly continuous is $X$ is compact? Moreover if $X$ has a finite number of isolated points, is $X$ compact now? I think that the answer to the ...
2
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1answer
45 views

$X $ is complete if every real valued continuous function on $X $ is uniformly continuous

If every real valued continuous function on $X $ is uniformly continuous,then is $X$ complete? My attempt:let $x_n$ be a Cauchy Sequence in $X$. Let $f$ be a real valued continuous function. To show ...
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1answer
46 views

show a subset of $\mathbb R^n$ is compact if it is closed and bounded

Use the two lemmas to prove that a subset of $\mathbb R^n$ is compact if it is closed and bounded. Lemma 1: A closed subset of a compact space is compact Lemma 2: If $X$ and $Y$ are compact then ...
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2answers
33 views

Compactness of two equivalent metric spaces

Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$). The following are ...
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1answer
104 views

What does it mean for a set to be compact? (intuitively)

I'm having trouble intuitively understanding what it means for a set to be compact. I know that by definition a set is compact if for every open cover of the set there exists a finite subcover. But I ...
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1answer
105 views

How can I prove that something is an open cover?

I want to prove that the union of some intervals forms an open cover for some segment. Any ideas on how to do this?
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2answers
90 views

Would a connected space contain a compact subspace

I am trying to prove that in a connected space - $X$ , for every two elements of $X$ - say $a,b$ I can find a subspace of $X$ ( say $X'$ ) , such that$ X'$ contains a,b and is also connected, and ...
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1answer
40 views

$C_b(X)$ is non-separable for $X$ non-compact

If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$. I was trying like this, but got stuck at middle: Take a ...
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1answer
42 views

Examples of non-compact connected spaces with the property…

I am looking for a non-compact connected space $X$ such that for any two disjoint closed $A,B\subseteq X$ there exists a proper closed connected $C\subseteq X$ such that $A\cup B\subseteq C$. I ...