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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compactly generated spaces

A topological space $X$ is called compactly generated if following condition holds: $A\subseteq X$ is open in $X$ iff for every compact $K\subseteq X$, $A\cap K$ is open in $K$. My lecturer said ...
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1answer
36 views

Maximum and minimum of a function from $\mathbb{R}^n$ to $\mathbb{R}$

Let $A \in \mathbb{R}^{n \times n}$ be a real $n \times n$-matrix. Consider the function $$q: \mathbb{R}^n \to \mathbb{R}, x \mapsto x^t A x$$ where $x^t$ is the transposed vector $x$. I now want ...
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1answer
17 views

unitalization of the $c^*$-algebra of complex polynoms without constant term / compactness of the spectrum of elements in non-unital $c^*$-algebras

Let A be $C^*$-algebra with unit $e$ and $a\in A$ normal. We define $$alg(a,a^*)=\overline{ \{ \sum\limits_{k,l=0}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$$ ...
4
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3answers
39 views

Prob 10 Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to give examples of this result failing?

Let $X$ be a compact topological space. Let $f_n \colon X \to \mathbb{R}$ be a sequence of continuous functions such that $f_n(x) \leq f_{n+1}(x)$ for all $x \in X$ and for all $n \in \mathbb{N}$. Let ...
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4answers
67 views

Are there compact manifolds without boundary?

Based on this question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the topology of the manifolds has to be the trace ...
3
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1answer
80 views

Prob 9, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to prove the generalised tube lemma?

The tube lemma is as follows: Let $X$ and $Y$ be topological spaces. Let $Y$ be compact. Let $x \in X$. If $N$ is an open set in $X \times Y$ such that $x \times Y \subset N$, then there is an open ...
3
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2answers
135 views

Complementary compactness

Let $X$ be a topological space having the property that whenever a subset $A$ of $X$ is compact, then $X\setminus A$ is compact too. Is every subset of $X$ compact?
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3answers
51 views

Compact Sets in $\mathbb{R^{n^2}}$ [duplicate]

I have a question of multivariable analysis and I don't know how to resolve this. The $n \times n$ orthogonal matrices form a compact subset of $\mathbb{R^{n^2}}$? I will be very grateful for ...
4
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1answer
42 views

Coincidence points on compact Hausdorff spaces.

I am really stuck on this exercise in my course notes. Let $X$ and $Y$ be compact Hausdorff spaces and $f, g : X \to Y$ be continuous functions. Show that: There is an $x \in X$ with $f(x) = ...
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1answer
25 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...
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1answer
137 views

Is this a compact space?

Let $A=\{x:d_\infty(x,0)\le 1 \}$, the subspace of the space of bounded sequences $x=(x_n)^\infty_{n=1}$, $x_n\in \mathbb{R}$, with metric $\{x:d_\infty(x,y)= sup_n |x_n-y_n| \}$. The answer says it ...
2
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0answers
106 views

If every real-valued continuous bounded function on a metric space $M$ attains its maximum (or minimum), then $M$ is compact

Suppose that $(M,d)$ is a metric space. I want to show if every continuous bounded function $f:M \rightarrow \mathbb{R}$ achieves a maximum or minimum, them $M$ is compact. I found a similar ...
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2answers
59 views

locally compact Hausdorff

A space $X$ is called locally compact if every point of $X$ has a compact neighbourhood. I want to show that If $X$ is Hausdorff then $X$ is locally compact iff for every $x$ of $X$, every ...
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0answers
18 views

every nonempty compact, locally path-connected and connected metric space is path-connected [duplicate]

I wanna prove that if $M$ is nonempty compact, locally path-connected and connected metric space then it is path connected. I think to prove this the best way is to show that between every to points ...
4
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2answers
57 views

Labelings of infinite directed acyclic graphs

Let $G=(V,E)$ be a countably infinite directed acyclic graph and $L$ be a finite set of vertex labels. The number $\left|V\right|$ of vertices is countable infinity and some vertices may have an ...
2
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1answer
78 views

Proofs of the Riesz–Markov–Kakutani representation theorem

Let $X$ be a compact Hausdorff space, $C(X)$ the set of all real continuous functions on $X$, and $\mathcal{B}$ be the Baire $\sigma$-algebra of $X$, which is the $\sigma$-algebra generated by the ...
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1answer
19 views

Sigma-compact Polish groups

I would like to see an example of a sigma-compact Polish group which is not locally compact. I know that e.g. $l^{\infty}$ is a topological group which is sigma-compact but not locally compact. But ...
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2answers
56 views

Examples about compactness

Compactness implies countably compactness which in turn implies limit-point compactness. Sequentially compactness implies limit point compactness. $Z_{+} \times \{0,1\}$ with two-point indiscrete ...
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1answer
42 views

Connected matrix Lie group

While enjoying Lie groups with Brian C. Hall's "Lie groups, Lie algebras, and representations", I'm stuck with the "standard argument using the compactness of the interval $[0,1]$" in the proof of the ...
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2answers
48 views

Confusion over the concept of “compactness”

I have to prove some stuff that involves the concept of collection, in particular those relating to compact sets. But then I have got this trouble. For example, consider the set of all rationals. If ...
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2answers
56 views

Construct an open cover of S with no finite subcover

Let S be a subset of Rn, and suppose that S is not bounded. Construct an open cover of S with no finite subcover, then prove this claim about your open cover. Let S be a subset of Rn such that S is ...
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1answer
28 views

Measure of open sets covering compact set

Prove that if $F$ is a finite collection of open intervals that covers a compact interval $[a, b]$, then the sum of the lengths of the intervals in the collection is strictly greater than $b − a$ ...
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2answers
77 views

Preserving compactness and connectedness implies continuity for functions between locally connected, locally compact spaces?

In this question: Connected and Compact preserving function is not continuous example? It is mentioned that "a function between locally-compact, locally-connected topological spaces which preserves ...
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1answer
41 views

About the weak compactness of a certain set.

Why is the following set weakly compact in $L^1(d\mu)$? $$\left\{-\frac{|x|^2}{2}+O(l)\right\}$$ where $\mu$ is a probability measure in $\mathbb{R}^n$ with finite second order moment: ...
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2answers
84 views

How to show there exists $E$ such that $E \cap K_n$ is dense for every $n$?

Let $\Omega$ be a region (nonempty connected open subset of the complex plane). Let $K_n$ be a sequence of compact sets whose union is $\Omega$, such that $K_n \subset \mathring{K_{n+1}}$ (the ...
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1answer
39 views

Convex open neighborhood of compact convex subset

I'm stuck on what ought to be a straightforward topology problem. Say $X$ is a compact convex subset of a locally convex space (everything in sight is assumed Hausdorff). Say $Y\subseteq X$ is a ...
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1answer
44 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
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1answer
35 views

generalize the question every every intersection of nested sequence of compact non-empty sets is compact and non-empty

I'm aware how to prove that the intersection of nested sequence of compact non-empty sets is compact and non-empty. but I want to generalize this question to transfer the hypothesis of having nested ...
4
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2answers
91 views

Quotient Maps and Compact Hausdroff Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X$ and $Y$ are compact Hausdroff space and $f:X\rightarrow Y$ is a continuous ...
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2answers
68 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
4
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2answers
49 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
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1answer
33 views

A sequentially compact subset of $\Bbb R^n$ is closed and bounded

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
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1answer
21 views

Is there a lower bound for the maximal number of separated sets?

Let $(X,d)$ be a metric space and $T\colon X\to X$ uniformly continuous. A set $E\subset X$ is said to be $(n,\varepsilon)$-separated if for any distinct $x,y\in E$ there is a $0\leq j< n$ such ...
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2answers
31 views

Compactness and Hausdorffness with different topology

Here is the question (Munkres pg. 170): Show that if $X$ is compact Hausdorff under both $\mathcal{T}$ and $\mathcal{T}'$, then either $\mathcal{T}$ and $\mathcal{T}'$ are equal or they are not ...
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1answer
26 views

Limit of bounded functions in compact-open topology

Let $(X, \mathscr{T})$ be a topological space and $(Y,d)$ be a metric space. Recall that the compact-open topology $\mathscr{T}_{co}$ on $Y^X$ is generated by the subbase $$ \mathscr{S} = \{ S(C, V) ...
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1answer
63 views

Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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1answer
34 views

Does $\sigma$ -compact imply separable?

Let $D$ be a metric space. If $D$ is $\sigma$-compact, does this imply that $D$ is separable? I thought I had a proof, but I think it is wrong. my proof: Let $K_n$ the compact sets such that $K_n ...
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2answers
380 views

Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
3
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1answer
30 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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0answers
47 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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0answers
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Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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1answer
56 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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1answer
33 views

Proving subsets of $l^{\infty}$ are compact

Recently I started reading up on some set theory and metric spaces. I just read about compact subsets and I thought I understood it but in the exercises I'm having difficulty with the following ...
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1answer
30 views

Is compact $T_1$ topological space hausdorff?

I'm in a middle of a very hard exercise which its goal is to prove that some space is hausdorff, but all I could show is that it is $T_1$. But I can also deduce that it is compact. Is that enough for ...
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1answer
26 views

Is Alexandroff duplicate compact?

Consider the Alexandroff duplicate $X\times_{ad} 2$, the space $X\times 2$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X$, $(U\times\{0,1\})\setminus (x,1)$ is ...
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2answers
106 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
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1answer
18 views

Bounded set that is not closed nor compact

I am to find a set that is bounded but not closed nor compact. Here are my ideas. Please tell me if any of my logic is flawed. I thank you in advance. Consider the set $A = (0,1)$ where $A \subset ...
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2answers
110 views

In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Prove that if $X$ is Hausdorff and $\mathfrak{C}$ is a nonempty chain of compact and connected subsets of $X$, then $\bigcap \mathfrak{C}$ is compact and connected. Here are the definitions which ...
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2answers
28 views

Negate a proposition with quantifier?

I'm going over the proof of the theorem stating that "In a metric space, compactness impliess sequential compactness". I'm very likely confusing myself. I have the following proposition: $\forall ...
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0answers
35 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...