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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2answers
50 views

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: ...
2
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0answers
59 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
6
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1answer
358 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
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2answers
60 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 ...
3
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3answers
48 views

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 ...
3
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2answers
39 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 ...
6
votes
1answer
191 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
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0answers
36 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 ...
1
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1answer
41 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 $X$ ...
7
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0answers
147 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 ...
0
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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 ...
1
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2answers
460 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
3
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2answers
27 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 ...
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0answers
41 views
4
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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
49 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 ...
1
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2answers
38 views

Let A and B be disjoint closed subsets of Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}. Show that if A={a} is a singleton, then d(A,B)>0.

Let $A$ and $B$ be disjoint closed subsets of $\mathbb{R}^n$. Define $d(A,B)=\inf \{||a-b||: a \in A, b \in B\}$. I have to show that if $A=\{a\}$ is a singleton set, then $d(A,B)>0$ and I have no ...
0
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1answer
43 views

Prove that the set of accumulation points of a bounded subset of $\mathbb{R}^n$ is compact.

We are allowed use of the Heine-Borel theorem that states that a set is compact in $\mathbb{R}^n$ iff the set is closed and bounded. I know that the set of accumulation points is closed, but I am ...
2
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1answer
65 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
1
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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)?
2
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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
56 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) ...
1
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1answer
76 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 ...
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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 ...
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 ...
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
27 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
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 ...
0
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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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0answers
26 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 ...
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0answers
19 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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0answers
25 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 ...
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0answers
31 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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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 ...
0
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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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3answers
48 views

Rudin - Exercise 12, Cap. 2 Principles of Mathematical Analysis

Let $\Bbb{K}\subset\Bbb{R}$ consist of $0$ and the numbers $\frac{1}{n}$, for $n=1,2,3,\dots$. Prove that $\Bbb{K}$ is compact directly from the definition (without using the Heine-Borel theorem).
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2answers
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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 ...
3
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2answers
92 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
48 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
29 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 ...
3
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1answer
82 views

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$.

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$. Context: $f \in \operatorname{Lip}_K[a,b]$ then it is ...
3
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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!
2
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3answers
122 views

Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis

I am not able to fill the gap in proof of following theorem which is stated as... Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an ...
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1answer
44 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
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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1answer
41 views
10
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4answers
4k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...