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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All neighborhoods of a compact subset of an open space are subsets of that open space

Let $K$ be a subset of $U$, with $K$ compact and $U$ open. Prove that there is an $\epsilon > 0$ such that for all $p$ in $K$, a neighborhood of radius $\epsilon$ of $p$ is a subset of $U$. Note: ...
3
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2answers
28 views

Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
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0answers
33 views

Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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1answer
36 views

$\mathbb{N}$ is a Compact Space with the Co-finite Topology?

Let $X$ be the topological space on the set $\mathbb{N}$ with the cofinite topology. I am having a hard time seeing why this is compact in the topological sense. If each open $n$-hood on $X$ ...
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1answer
40 views

A compact set, which is not closed.

I'm looking for a compact set, which is not closed. I read somewhere that $Z^+$ are compact and not closed, but I don't understand why. Are there any other examples of compact sets that are not ...
7
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1answer
132 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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0answers
40 views

If a set is closed, why is that set intersected with a compact set closed?

If F is a closed subset of K and K is compact, why is F intersect K closed?
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1answer
48 views

Baire sets in locally compact Hausdorff spaces

(This is a follow-up to Compact $G_\delta$ subsets of locally compact Hausdorff spaces.) Suppose $X$ is a locally compact Hausdorff space. The Baire sets in $X$, denoted by $\mathcal Ba(X)$, comprise ...
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0answers
55 views

Conjecture in continuum theory: my proof attempt

Conjecture. Suppose $X$ is a normal connected space such that every nondegenerate closed subset of $X$ is disconnected. Then every proper subcontinuum of $\beta X$ has empty interior. proof attempt. ...
2
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0answers
47 views

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a ...
2
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1answer
64 views

A direct proof that a compact metric space is sequentially compact

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks
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1answer
27 views

Are pseudocompact metric spaces complete?

Is there a way to show that pseudocompactness on a metric space implies completeness directly (without using sequential compactness)?
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1answer
43 views

Compact sets closed in Hausdorff spaces without choice?

An elementary proof that compact sets are closed in Hausdorff spaces involves making arbitrary choices based on the Hausdorff property. Is there a way to avoid invoking choice?
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0answers
38 views

If (X,d) is a separable metric space then there exists a metric d′ that is topologically equivalent to d and such that (X,d′) is totally bounded.

I know that this question Separability, total boundness and topological equivalence of metrics has been asked, but the only solution given is not valid. There is something I already knew: (Y, d2) ...
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2answers
24 views

Compact zero-dimensional $T_2$-topologies on $\mathbb{N}$

Let $\tau$ be a compact topology on $\mathbb{N}$ such that for every two points $m\neq n\in \mathbb{N}$ there is a clopen set $U$ containing $m$ but not $n$. Is $(\mathbb{N},\tau)$ isomorphic to ...
1
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1answer
58 views

Separability, total boundness and topological equivalence of metrics

The problem is: If $(X,d)$ is a separable metric space then there exists a metric $d'$ that is topologically equivalent to $d$ and such that $(X,d')$ is totally bounded. I know that if ...
3
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1answer
50 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
40 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
35 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
42 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 ...
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1answer
26 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 ...
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0answers
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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 ...
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1answer
85 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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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 ...
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3answers
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Problem 2.5.10 in Kreyszig's Functional Analysis Book

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
30 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 ...
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0answers
18 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 ...
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1answer
42 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
38 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
35 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 ...
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2answers
91 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 ...
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1answer
47 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 ...
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1answer
28 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 ...
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1answer
15 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 ...
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0answers
17 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
43 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
40 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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66 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 ...
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1answer
44 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
21 views

Let $X$ be a compact Hausdorff space, then $X$ is second countable if and only if $C(X)$ is separable.

All is in the title: let $X$ be a compact Hausdorff space, then $X$ is second countable if and only if $C(X)$ is separable. I know this is a well known theorem, it is mentioned in several other posts ...
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0answers
27 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
36 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
34 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) ...
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1answer
36 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 ...
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1answer
21 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 ...
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0answers
85 views

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

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
46 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 ...
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1answer
50 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
31 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
40 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, ...