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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3answers
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Is the $\omega$-product of the set of irrationals compact?

We know that any product of compact spaces is compact. But, I wonder that the countable product of $\mathbb{P}$ can be compact since $\mathbb{P}$ is not compact?
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3answers
62 views

Compact subsets of a metric space

I am trying to to prove that f: X --> Y is continuous on X if and only if f is continuous on every compact subset of X. X and Y are metric spaces. How do I show that every point of X belongs to some ...
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1answer
309 views

Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...
3
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0answers
79 views

References for the Čech-Stone compactification of Hyper-Reals?

It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when ...
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2answers
98 views

Compactness in Infinite Dimensional Vector Spaces

Show that, in an infinite dimensional normed space $(V,\|\cdot\|)$, the closed ball of radius $2$ $$ B_2:=\{x\in V:\ \|x\|\leq2\} $$ is not compact. I suspect I am not understanding what is going ...
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1answer
69 views

Show that these subsets of R are sequentially compact

I have to show that a) $[2, 2\frac{1}{2}] \cup [3, 3\frac{1}{3}] \cup [4, 4\frac{1}{4}] \cup ...$ b){1, 2, 3, ..., $N$} for some $ N \in \mathbb{N} $ are sequentially compact. I know that in a ...
2
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1answer
46 views

Does the union of all these neighborhood cover $[0,1]$

Consider the rationals in $[0,1]$. Around each I take a neighborhood (possibly of different radii). Is the union of all these neighborhood sure to cover $[0,1]$? What if I had used irrationals instead ...
3
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2answers
126 views

Completeness/Compactness of a subset in a normed linear space

Let $(X,\|\cdot\|)$ be the normed linear space consisting of the sequences $a=(a_n)_{n=1}^{\infty}$, for which the corresponding series $\sum_{n=1}^{\infty} a_n$ converges absolutely, with norm ...
2
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6answers
73 views

Show that $A= ([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q $ is not compact.

We have $\Bbb Q$ equipped with the Euclidean Metric. Show that $A=([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q $ is not compact. How would you go about showing this? You can make on open cover ...
2
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2answers
174 views

Topology Question: Compact Hausdorff Spaces

If T and T′ are two distinct compact Hausdorff topologies on X, what can we say about (X,T∩T′) (we know that T∩T′ is a topology on X). Is it compact? Is it Hausdorff?
2
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1answer
392 views

One point compactification of the rationals is not second countable

As a counter example to show that second countable is a property which is not preserved under a continuous function, I consider the rationals under the discrete topology, which is second countable, ...
3
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0answers
77 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
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0answers
74 views

Analytic function on a compact set

I am trying to prove the following but I am not sure how to use the compactness: Let $f$ be a real analytic function on a compact set $K$. Let $(x_n)_n$ be a sequence in $K$ such that $f(x_n) = 0$. ...
4
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1answer
129 views

Algebra of Functions and Compactification

Let $X$ be a completely regular topological space. Let $\mathscr F$ be a family of continuous functions mapping from $X$ to $[0,1]$ that separates points from closed sets, i.e., whenever $C\subseteq ...
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6answers
869 views

Why is an open interval not a compact set?

I learned that every compact set is closed and bounded; and also that an open set is usually not compact. How to show that a concrete open set, for example the interval $(0,1)$, is not compact? I ...
1
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2answers
55 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
2
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1answer
144 views

Arzela-Ascoli net question

Let $X$ be a compact metric space. Let $C(X)$ denote the space of real-valued continuous functions on $X$. A commonly given corollary to the Arzela-Ascoli theorem is: Proposition: If $f_n$ is an ...
4
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1answer
62 views

Example of topological space where pseudo-component differ with intersection of clopen sets.

It is well known fact that connected component $C_x$ of a point $x$ from some topological space $\tau$ is contained in every clopen set containing $x$ (so it's intersection $M$ also contains $C_x$). ...
2
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5answers
207 views

Real Analysis: Compact Sets

I'm working on a general real analysis problem involving compact sets. I was given these two sets: $$A = \left\{0, 1, \frac{1}{2}, \frac{1}{3}, \dots , \frac{1}{n}, \dots\right\}\text{ and } B = ...
1
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2answers
45 views

Negative exponential distance

Let $X := \left\{(a_k)_{k \in \mathbb N}, a_k \in \mathbb C\right\}$. Let $d\left( (a_k)_{k \in \mathbb N}, (b_k)_{k \in \mathbb N} \right) := e^{-u}$ with $u$ the smallest integer $k$ such that $a_k ...
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1answer
76 views

How prove that $\mathbb{CP}^2$ is compact? [closed]

How prove that $\mathbb{CP}^2$ is a compact manifold.
3
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1answer
75 views

Continuous function and nested compact spaces

Let $X,Y$ be metric spaces and $f:X \to Y$ be a continuous function. Let $K_n \subset X$ be a compact subspace of $X$ for $n \in \mathbb N$ such that $K_{n+1} \subset K_n$. Prove that $f(\bigcap_{n ...
5
votes
2answers
192 views

Topology problem - compactness

How to solve the following: Let $X$ be a locally compact, $Y$ Hausdorff space and $f : X\rightarrow Y$ continuous open surjection. Prove that for every compact set $K\subset Y$ exists compact set ...
2
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1answer
96 views

Equivalence Relation, Quotient Space, Compactness

Define an equivalence relation on $\mathbb{R}^2$ by $a \times b$ ~ $c \times d$ iff $a-c$ and $b-d$ are both integers and let $T=(\mathbb{R}^2)^*$ denote the corresponding quotient space. (a) Show ...
0
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1answer
60 views

Uniformly continuous on a compact set, still uniform on a subset?

So if I have a function that is uniformly continuous on a compact set K, do all subsets of K inherit the uniform continuity? If I restrict myself to the reals, this seems to be true. But what happens ...
0
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1answer
62 views

Compact subsets of $c_0$

Let $c_0$ be the Banach space of all sequences converging to 0, equipped with the supremum norm. How do the compact subsets of $c_0$ look like? I could imagine that $K \subset c_0$ is compact if ...
3
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3answers
120 views

How to show that a Stone Čech compactification of $\mathbb Z_+$ has a cardinality at least $2^\frak c$

Show that $\beta\mathbb Z_+$ (The Stone Čech compactification of the positive integers) has cardinality at least as great as $I^I$ where $I=[0,1]$. I know that I is compact Hausdorff and so $I^I$ is ...
2
votes
1answer
330 views

Is $\mathbb{R}^\omega$ locally compact in the box topology?

I know that $\mathbb{R}^\omega$ is not locally compact in the product topology. Since the box topology is finer than the product topology, does that mean that $\mathbb{R}^\omega$ is not locally ...
3
votes
2answers
497 views

Quasicomponents and components in compact Hausdorff space

Let $X$ be a compact Hausdorff space, $x,y\in X$ and $\mathcal{A}$ a colection of closed subspaces of $X$ such that for every $A\in \mathcal{A}$ then $x$ and $y$ are in the same quasicomponent of $A$. ...
3
votes
1answer
97 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
2
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1answer
48 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
3
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2answers
93 views

Compactness of $x^4+y^4=1$

Show $A=${$(x,y):x^4+y^4=1$} is compact. So far, I'm thinking I should mention that $0 \le x \le y-x \le 1$ and $0 \le y \le x-y \le 1$ determines the values of $(x,y)$, and since [0,1] is compact, ...
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2answers
143 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding this ...
3
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1answer
165 views

Pictures of One-point compactification

Are these the pictures of the following one-point compactification of the following surfaces: Two dimensional sphere with three points removed The disjoint union of two copies of $S^{1} \times ...
2
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1answer
48 views

Compactness of $\operatorname{Proj}S$

It is well-known that for any ring $A$, $\operatorname{Spec}A$ is quasi-compact. Is it true in general that $\operatorname{Proj}S$ is quasi-compact, where $S$ is an $\mathbb{N}$-graded ring? ...
2
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1answer
45 views

Cardinality of a Rosenthal compact space?

Let $X$ be a polish space a real valued function $f$ on $X$ is of the first Baire class if $f$ is a pointwise limit of a sequence of continuous functions on $X$. Let $B_{1}(X)$ denote the space of all ...
0
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1answer
46 views

Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
4
votes
2answers
91 views

Show that A=$\{(x_1,…x_n) \in \Bbb R | -1\le x_1\le x_2\le …x_n\le 1\} \subset \Bbb R^n $ is closed.

The full question was: Show that A=$\{(x_1,...x_n) \in \Bbb R | -1\le x_1\le x_2\le ...x_n\le 1\} \subset \Bbb R^n $ is compact, but I was able to show correctly that it is bounded. However my ...
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3answers
332 views

How to show the intersection of two compact subsets is compact

Let (X,d) be a metric space and A,B $\subset$ X be two compact subsets. Show $A\cap B$ is also compact. I attempted this question by showing the intersection is bounded and closed. But I stated ...
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2answers
44 views

how can i prove that the sorgenfrey line is not sigma compact?

Can someone give me a hint to prove that the sorgenfrey line is not sigma compact? thanks in advance
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0answers
52 views

Quasicompact? Why the distinction?

What is the reason that some topologists use quasicompact? Why is the distinction made? quasi means "not really", so why use this terminology?
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2answers
33 views

Compactness of a set in a Locally compact Hausdorff space

My question is about the following lemma: Let $X$ be a locally compact Hausdorff space, and $V$ be a nbhd. of a point $x \in X$. Then there is a nbhd. $U_{x}$ of $x$ such that $\overline{U_{x}}$ is a ...
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1answer
402 views

$X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
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2answers
78 views

the product $(X_1, T_1)\times\ldots\times(X_n,T_n)$ is locally compact, then each $(X_i, T_i)$ is locally compact.

Prove that the product $(X_1, T_1)\times\ldots\times(X_n, T_n)$ is locally compact, then each $(X_i, T_i)$ is locally compact. My proof: Let $p=\langle x_1,\ldots,x_n\rangle$ be in the product. ...
0
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3answers
120 views

the product of topological spaces is compact, then each of its factors is compact.

Let (X, T1) and (Y,T2) be topological spaces. Prove that if (X1 x X2, T) is compact, then each (X, T1), (Y,T2) is compact. Here is how I start this problem: Since (X x Y, T) is compact, each of its ...
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1answer
48 views

Demonstrate that the following metric space is not compact

Let $X$ be a metric space. Show, if there is an $r > 0$ and a sequence $(x_n)$ from $X$ such that $d(x_n,x_m) \geqslant r$ for $n≠m$, then $X$ is not compact. I know that sequentially compact and ...
3
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2answers
51 views

showing compactness for a subset of a function space

Our professor told us the following in lecture: Let $X_A:=\{f\colon A\to\mathbb R|f(A) \textrm{ is bounded}\}$ and $\alpha(f,g):=\sup\{|f(x)-g(x)| \;|x\in A\}$. Given $\beta\colon A\to\mathbb R, ...
0
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1answer
67 views

Motivation for a proof “In a regular space, if every open cover contains a countably locally finite open refinement, then the space is paracompact”.

Let $(X,\tau)$ be a regular topological space. It's a theorem that the followings are equivalent: (1) Every open cover has a countably locally finite open refinement. (2) Every open cover has a ...
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1answer
59 views

Sorgenfrey's topology- compact subsets

Consider the topological space $(R,\tau_{sor})$ Are the atoms $\{x\}$ where $x\in X $ compact in Sorgenfrey's topology? And in the usual topology? To say if $\{0\}\times [0,\infty)$ is compact in ...
4
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1answer
98 views

Metrizable topological space $X$ with every admissible metric complete then $X$ is compact

How to prove: If $X$ is a metrizable topological space and every admissible metric on $X$ is complete then $X$ is compact. I was trying with an idea of contradiction and thereby to construct ...