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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Three questions from σ-compact spaces and topological groups

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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20 views

Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Let $X$ be an ordered set in which every closed interval is compact. Then $X$ has the least upper bound property. How to prove this? My effort: Let $A$ be a non-empty subset of $X$ such that $A$ is ...
2
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2answers
49 views

Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
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1answer
17 views

Proof cube & trapezium a Compact Space & $E^n$ & $I^n, I^{\infty}$ are connected space ??? [on hold]

I need a serious help here please! Question 1: Prove that $E^n$ & $I^n$, $I^{\infty}$ are connected spaces. After a lot of search I found some two theorems in James Dugundji book. But I still ...
6
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3answers
62 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact, and let $f \colon X \to Y$ be a closed, surjective, and continuous map such that, for each $y \in Y$, the inverse image $f^{-1} ( \ \{y ...
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1answer
15 views

Additive function in $\mathbb{R}^n$ is continuous, and related subspaces compact

I want to show that the function: $A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$ is continuous. Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, ...
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1answer
26 views

Connection between one-point compactification of $\mathbb{R}$ and $S^1$

Definition I got: A one-point compactification of $X$ is $\hat{X}=(X\cup\{\infty\},\tau).$ The new topology $\tau$ is generated by open subsets of $X$, and $U\cup\{\infty\}$ where $U=K^C$ where $K$ ...
2
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1answer
40 views

Is there an example of a non compact operator whose square is compact?

Is there an example of a non compact linear operator T from a Banach space X to itself such that T^2 is compact? Of course the converse is true, as T ^2 is compact if T is. Here T^2 means T composite ...
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1answer
35 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
2
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1answer
24 views

What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other?

Let $\tau ,\tau_1$ be two topologies on the set $\mathbb R$ .Suppose $\tau \subset \tau_1$ .What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other? ...
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1answer
44 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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0answers
31 views

A confusion of a real analysis online lecture: Relative compactness

https://www.youtube.com/watch?v=kkKfRaI-cqs At 13.00 what does the professor mean to let those subcovers be "restricted" to Y? Is that a process like A is contained by B implies that A intersect C in ...
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0answers
12 views

core-compact but not locally copact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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2answers
32 views

continuous image of a locally compact space is locally compact

Is continuous image of a locally compact space is locally compact? Let $X$ be locally compact(l.c.).Let $f:X\to Y$ is continuous and surjective. A space $X$ is locally compact if for each $x\in X$ ...
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2answers
43 views

compactness of Hilbert cube

I want to show that the Hilbert cube which is: $H=\{(x_1,x_2,...) \in [0,1]^{\infty} : for \ each \ n \in \mathbb{N}, |x_n|\leq \dfrac{1}{2^n}\}$ is compact with respect to the metric: ...
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0answers
25 views

Prob. 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How is the projection onto the first factor closed in the second factor is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact. Then how to show that the projection map $\pi_1 \colon X \times Y \to X$ is a closed map? My effort: Let $C$ be a non-empty closed ...
2
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1answer
30 views

Arbitrary intersection of closed compact sets is compact (Topology)

Arbitrary intersection of closed compact sets is compact We've been trying to find a counter example to this, however we failed. So we would be happy if someone can tell us if this proposition is ...
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1answer
13 views

Construction of a given neighbourhood in a locally compact group

Let $G$ be a locally compact group. Why is it possible to select a compact neighbourhood $U$ of $e \in G$ such that $U=U^{-1}$ and $gU^2 \subset V$? This is a construction quickly stated by Helgason, ...
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2answers
31 views

Product of Compactly Generated and Locally Compact is Compactly Generated

I'm trying to prove that if $X$ is compactly generated and $Y$ is T2 (Hausdorff) and locally compact then $X\times Y$ is compactly generated. First it is clear that since both $X$ and $Y$ are T2 then ...
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1answer
30 views

Let $K_1 \supset K_2 \supset… $ be a sequence of connected compact subsets of $ \Bbb R^2 $. Is $ K = \cap_{i=1}^\infty K_i $ is connected? [duplicate]

I have managed to write down two proofs showing the connectedness of $K$. But still shaky about both of them. Here are the proofs: 1)Suppose $K$ is disconnected. Then we write it's separation as $ K ...
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1answer
86 views

Epimorphisms of locally compact spaces

Let $LCH$ be the category of locally compact Hausdorff spaces with proper continuous maps. Question. What are the epimorphisms in $LCH$? I suspect them to be surjective, but I haven't been able to ...
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15 views

About submanifolds, diffeomorphisms, compact subsets and jordan measurable subsets.

I am given the following problem set which left me pretty puzzled. Let $M \subset \mathbb{R}^n$ be a $C^1$ submanifold and $$\psi : \Omega \rightarrow U \cap M$$ a chart of $M$. $\Omega \subset ...
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4answers
37 views

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics: $1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$ $2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$ My try: In order to ...
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0answers
22 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
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3answers
44 views

Topology: Continuous bijective function, domain = covering compact

Final in real analysis coming up. I could really use some help. If a function f from one set M to another set N is a continuous bijection and M is covering compact, can anything in general be said ...
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1answer
27 views

function on k-space

A topological space $X$ is called k-space if the following condition holds: $A\subseteq X$ is open in $X\iff A\cap K$ is open in $K$ for any compact subest $K$ of $X$. A space $kX$ is a topological ...
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1answer
32 views

Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$

Let $A,B$ be two compact subsets of $X$ where $(X,d)$ is a metric space. 1.Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$ where $d(A,B)=\sup\{d(a,b):a \in A;b\in B\}$ 2.Show that ...
4
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1answer
22 views

Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
2
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1answer
32 views

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
34 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
14 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 $$alg(a,a^*)=\overline{ \{ \sum\limits_{k,l=0}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$$ ...
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3answers
37 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
53 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 ...
2
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1answer
58 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 ...
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2answers
131 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
50 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
41 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
21 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
134 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 ...
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0answers
64 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
54 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
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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 ...
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2answers
54 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 ...
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1answer
39 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
10 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
46 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
38 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
46 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
51 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
25 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 $b − a$ ...