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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Compactness Theorem (Propositional Logic) and Compactness (Metric spaces). [duplicate]

Definition. A subset $E$ of a metric space $(X,\tau)$ is compact if every open cover of $E$ has a finite subcover. Theorem (Compactness Theorem). A set $\Gamma$ of formulas is ...
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1answer
23 views

infinite subset of discrete metric space is not compact

The question is Im not really sure how to go about this So far i am trying to show that for an open cover of the infinite subset X, there isn't a finite sub cover and therefore X is not compact I ...
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1answer
23 views

Weierstrass theorem and compactness

On my book the statement of Weierstrass theorem is: If $f$ is a continuous function $f:A\subseteq X\rightarrow \mathbb{R}$ defined on a compact set $C$, where $A$ is the domain of $f$ and $X$ is a ...
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1answer
34 views

Describing a one-point compactification

Show that $X = S^{1} \cup ((0,2) \times \{0\}) \subset \mathbb{R}^{2}$ is locally compact and find its one-point compactification. Definitions $S^{1}$ is the open unit circle. Our definitions are ...
3
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1answer
81 views

1-point compactification and embedding

I'm totally stuck with the following question, I even don't know how to start: For each natural number $n$ we consider a space $X_n$ that is obtained by removing $n$ distinct points from ...
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0answers
28 views

$\overline{\mathrm{conv}} \{x_i : i \in \mathbf{N} \}$ is compact if $x_i \rightarrow 0$

I want to show that the set $\overline{\mathrm{conv}} \{x_i : i \in \mathbf{N} \}$ is compact in a Banach-space $X$ if $(x_i)_{i \in \mathbf{N}}$ is a sequence in $X$ converging to the origin. My ...
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0answers
10 views

Criterion for relative compactness in uniform spaces

I am having problems in understanding a criterion for relative compactness given in a book (see below for details if you are interested) on SPDEs. However, I think it just invokes a pretty general ...
2
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2answers
43 views

When is the image of a proper map closed?

A map is called proper if the pre-image of a compact set is again compact. In the Differential Forms in Algebraic Topology by Bott and Tu, they remark that the image of a proper map $f: \mathbb{R}^n ...
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75 views

Show precompactness for not necessarily continuous functions

I know that for showing precompactness of subsets of continuous functions, Arzelà-Ascoli is the tool of choice. However, the setting I'm facing here are functions in $L^1$. More precisely: Let ...
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1answer
32 views

Closed, continuous, surjective map with inverse images compact

Old qual question: Let $p:X\to Y$ be a closed, continuous, surjective map such that $p^{-1}(y)$ is compact for every $y\in Y$. Let $(U_\alpha)_{\alpha\in A}$ be an open cover of $X$. Show that any ...
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2answers
48 views

lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$ in Metric Space - Implications

A Metric Space $<M,d>$ is given by the Metric $M$ and distance function $d$ If there exists a Cauchy Sequence $x_n$ such that: lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$, for some $a \in ...
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2answers
20 views

if A is a compact subset of X,the inter section of all open sets which include A is a compact subset of X

I want to show that if A is a compact subset of X,the inter section of all open sets which include A is a compact subset of X,but intersection of all closed sets which include A is not necessarily ...
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0answers
56 views

How to understand intuition behind compactness? [duplicate]

I have taken a course in general topology this semester.while solving problems,i find it difficult to go by the definition which says that a space is compact if every open cover of it has a finite ...
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1answer
43 views

$0<\beta < \alpha \leq 1$, unit ball of Hölder space $C^{0,\alpha}[0,1]$ compact in $C^{0,\beta}[0,1]$?

So this is a very basic question on Hölder spaces. Let $0 < \beta < \alpha \leq 1$. Prove that the unit ball of $C^{0,\alpha}[0,1]$ is compact in $C^{0,\beta}[0,1]$. For reference: $\| ...
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1answer
32 views

Locally compact Haussdorff space

Theorem 29.2. Let $X$ be a Hausdorff space. Then $X$ is locally compact if and only if given $x \in X$, and given a neighborhood $U$ of $x$,there is a neighborhood $V$ of $x$ such that $\bar{V}$ is ...
2
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3answers
50 views

Creating a nested sequence of compact subsets of an open set in $\mathbb{R}$

Take $U \subset \mathbb{R}$ be an open set. I want to show that there exists nested compact sets $C_1 \subset C_2 \subset \dots$ such that $\bigcup_{j=1}^{\infty} C_j = U$. I have a feeling that I ...
0
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1answer
41 views

The union of a set with its complementary components that have compact closure [closed]

Let $X$ be a topological space and $A$ be a subset of $X$. Let $A'$ be a subset of $X$ such that $A'$ contains $A$ and all those subsets of $X\setminus A$ which are maximal connected sets and ...
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0answers
16 views

Prove multivariable function is onto

Suppose $f:\mathbb{R}^n\to\mathbb{R}^n$ is a continuously differentiable function such that the total derivative $Df$ is invertible at every point, and $f^{-1}(K)$ is compact for every compact ...
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0answers
55 views

If we think of infinity as a number, how does it affect the compactness/completeness of a metric space?

I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. ...
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3answers
33 views

Range of continuous function related

For a continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$,let $Z(f)=\{x\in \mathbb{R}:f(x)=0 \}$. Then which of the following is true for $Z(f)$ $Z(f)$ is always open. $Z(f)$ is always ...
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2answers
33 views

compact metric spaces and infimum

I am currently revising metric spaces and have come across a question which I am unable to answer and have no idea how to begin with. Let $(M,d)$ be a compact metric space. Suppose $T \colon M \to ...
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2answers
41 views

If $\hat A= A \cup \left\{\right.$ connected components of $X-A$ which are relatively compact in $X\left.\right\}$, then for every $A \subseteq X$

(Here, $B$ is relatively compact means the closure of $B$ is compact.) $\hat A$ is compact. $\hat A=\hat {\hat A}$. $\hat A$ is connected. $\hat A=X$. I try to eliminate the options ...
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1answer
40 views

Show that $|f(x)-f(y)| < M|x-y| + \epsilon$ for all $x, y \in X$

Let $X$ be compact subset of $\Bbb R $ and the function $f : X \to \Bbb R$ is continuous then given $ \epsilon > 0$ show that there is $M>0$ such that for all $x, y \in X$ $ |f(x)-f(y)| \le ...
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4answers
36 views

Finite union of compact sets is compact using a different def of open cover

I am trying to prove from definition that a finite union of compact sets is compact given that the definition of an open cover I have from my lecture notes is: An open cover $\cal U$ of a space $M$ ...
3
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1answer
84 views

Determining compactness and completeness of metric space

Metric Space: (M,d) Set: $M = \{ (x,y)\in \mathbb{R}^2:y>0 $ or $ x=0=y \}$ Metric: $d((x,y),(a,b)) = $min$\{ $max$ \{ |x-a|,|y-b| \},y+b \}$ $\space$ completeness: $\lim_{n ...
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1answer
55 views

Topology on $\mathbb{Z}$ [closed]

Consider the set $\mathbb{Z}$ of integers,with the topology $\tau$ in which every set is closed if and only if it is empty or $\mathbb{Z}$ or finite. Then which of the following statements are true? ...
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2answers
52 views

Difference between completeness and compactness

According to Wikipedia: A metric space $M$ is said to be complete if every Cauchy sequence converges in $M$ $ $ A metric space $M$ is compact if every sequence in $M$ has a subsequence ...
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3answers
42 views

Definition of compactness of metric spaces

In my lecturer's notes it says the following: Let $A \subseteq M$ and let $B = \{U_i : i \in I\}$ be an open cover of $A$. When determining the compactness or not of $A$, we might question whether it ...
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1answer
27 views

Topological Space With Matrix Elements

Let $U$ denote the set of all $n\times n$ matrices $A$ with complex entries such that $A$ is unitary(i.e $A^* A=I_n$). Then $U$ is a topological subspace of $C^{n^2}$,then which of the following is ...
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3answers
93 views

Subset of $\ell^2$ is precompact

Suppose we have a sequence of $a_i$ with some restrictions on it. Which restrictions must be to make set $$A= \left\{(x_i) \in \ell_2 \mid \sum\limits_{i\geqslant1} |a_i x_i|^2 \leqslant 1 \right\} ...
2
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1answer
59 views

Open compactification of metric space

Suppose I have a separable metric space $X$. I wanted to ask if there exists a separable metric compactification of this space $\overline{X}$, s.t. $X$ is considered open in $\overline{X}$.
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2answers
38 views

Compactness of a subspace of $L^{2}([0,1])$

Let's consider the following space $K \subset L^{2}[0, 1]$, consisting of fucntions $x(t)$ so that $\sin{t} \leq x(t) \leq t$. How to check, whether this subspace is compact in $L^{2}[0, 1]$ or not. ...
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17 views

Show $\mathbb{R}^\omega$ is not locally compact in product topology [duplicate]

Show that $\mathbb{R}^\omega$ is locally compact in the product topology.
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2answers
70 views

The set $X = \{0\} \cup \{\frac{1}{n} : n \in \mathbb{Z^{+}} \}$ is compact

I am reading the section on compactness from Munkres their is a certain part I don't understand. Consider the following subspace of $\Bbb R$: $X = \{0\} \cup \{\frac{1}{n} : n \in \mathbb{Z^{+}} \}$ ...
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3answers
65 views

Compact surface

To check if the surface $x^2-y^2+z^4=1$ is compact, we have to check if the surface is closed and bounded. Could you give me some hints how exactly we check that? How can we check if it closed and ...
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1answer
45 views

Find on $C[0,1]$ closed and bounded set $A$ that there are no such $f,g \in A$ that imply $\operatorname{diam}(A)=d(f,g)$

Find in $C[0,1]$ closed and bounded set $A$ such that there are no $f,g \in A$ that imply $\operatorname{diam}(A)=d(f,g)$, where $\operatorname{diam}(A) = \sup\{d(f,g)\mid f,g \in A\}$.
2
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1answer
74 views

Which of the following sets are compact:

Which of the following sets are compact: $\{(x,y,z)\in \Bbb R^3:x^2+y^2+z^2=1\}$ in the Euclidean topology. $\{(z_1,z_2,z_3)\in \Bbb C^3:{z_1}^2+{z_2}^2+{z_3}^2=1\}$ in the Euclidean topology. ...
2
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1answer
41 views

Easy examples of the Arzela-Ascoli Theorem

Let $X$ be a compact metric space. $M \subseteq C(X)$ is relatively compact if and only if $M$ (i.e. its elements) is equicontinuous and uniformly bounded. I've been told that this theorem gives ...
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2answers
30 views

A locally compact Hausdorff space is compactly generated

I am having trouble showing one direction of the proof that a locally compact Hausdorff space is compactly generated. Specifically, my question is how do I show that: if X is a locally compact ...
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1answer
91 views

Compact subset in colimit of spaces

I found at the beginning of tom Dieck's Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ ...
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2answers
160 views

Why does this proof fail?

I'm reading some notes on topology, and the notes' author is trying to raise motivation to consider compactness by providing a theorem whose proof is built intentionally wrong, but I don't agree with ...
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2answers
66 views

Which of the Following Sets are compact (C.S.I.R 2015)

$\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 1 \}$ in the Euclidean Topology $\{ (z_1,z_2,z_3) \in \mathbb C^3 : z_1^2 + z_2^2 + z_3^2 = 1 \}$ in the Euclidean Topology. $\prod_{n=1}^{\infty} A_n$ ...
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3answers
93 views

Importance of Locally Compact Hausdorff Spaces

I mostly deal with measure and probability theory and quite often, whenever I look up something on wikipedia, I see the mathematical objects defined on a locally compact Hausdorff space. I have very ...
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1answer
74 views

compactness of sets in euclidean topology and product topology

Which sets are compact in euclidean topology and product topology ? $\{(z_1,z_2,z_3):z_1^2+z_2^2+z_3^2=1)\}$ in the euclidean topology. $\{z\in \mathbb C:|re(z)|\leq a\}$ in the euclidean topology ...
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1answer
51 views

on existence of supremum/infimum

It is known that minimum or maximum of a function does not always exist but the supremum/infimum usually tends to exist. Example 1: For example, if we consider $X$ as the set or rational numbers with ...
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1answer
43 views

Intermediate Value Theorem on $\mathbb{R^n}$

Let $S^2$ denotes the subset of $\mathbb{R^3}$ which includes the points $(x,y,z)$ s.t $x^2+y^2+z^2=1$ i.e the boundary of a unit sphere. Let $f$ be a continuous function from $S^2$ to $\mathbb{R}$ ...
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1answer
46 views

Compact subset of a non compact topological space

Define a topological space X that is not compact and define a set A ⊂ X that is compact. Use the definition of finite open subcovers to show that A is compact. Ok so I think that a topological space ...
7
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2answers
120 views

Intuition for Kuratowski-Mrowka characterization of compactness

Fact. A space $X$ is compact iff for every space $Y$, the projection $X\times Y\rightarrow Y$ is a closed map. The finite subcover definition of compactness seems reasonably intuitive: finite covers ...
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3answers
54 views

Continuity of improper integral with a continuous integrand.

I am a newbie in analysis and am trying to wrap my head around some continuity/compactness/finiteness concepts. Let $f(x,y):\mathbb{R}^2\mapsto\mathbb{R}$ be a continuous function in both $x$ and $y$ ...
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2answers
54 views

$X$ compact Hausdorff with $X=X_1\cup X_2$. If $X_1,X_2$ are closed and metrizable, show that $X$ is metrizable.

This is Exercise 9 from Section 34 of Munkres - Topology. Following the hint given, I've done the following:Since $X$ is compact, $X_1,X_2$ are compact metrizable and hence have countable bases. Let ...