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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A problem in locally compact Hausdorff space

I am trying to solve the following problem. Let $X$ be locally compact Hausdorff and $Y$ be Hausdorff. (a) If $f: X \to Y$ is continuous and open map then show that $f(X)$ is locally compact. (b) ...
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Which of the following are compact?

Which of the following are compact? $1$.the set of all upper triangular matrices all of whose eigen values satisfy $|\lambda|\leq 2$ $2.$the set of all real symmetric matrices all of whose eigen ...
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Example of continuous functions $f\colon S \to T$ such that $f(S)=T$.

I would like to find an example of a continuous function from $S=(0,1)$ to $T=(0,1)\cup (1,2)$ such that $f(S)=(T)$. At the moment the only thing I can think might work would be to check whether ...
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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 ...
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Let (X, d) be a metric space and A, B ⊂ X be two compact subsets. Show that A ∩ B is also compact

Question seems fine i just have a few doubts. Is it possible to just use the Heine Borel theorem? as both A and B are compact it implies they are both closed, so therefore their intersection is ...
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35 views

A set of real numbers whose limit points from a countable set

Construct a set of real numbers whose limit points from a countable set. Is the set you constructed closed? Is it compact? My example is $$G=\{1/n+1/m: n, m \in \mathbb N\}\cup \{0\}$$ and as ...
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Show compactness of a set given by inequalities

Show that the subset $A=\{(x_1,...,x_n)\in\Bbb R^n |−1≤x_1 ≤x_2 ≤···≤x_n ≤1\}$ is compact. A is contain in an open cover as it is contained in $\Bbb R^n$. Therefore there exists a finite sub cover ...
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Characterising the discrete topology with compact subsets [duplicate]

If a set is endowed with the discrete topology then a subset is compact iff it is finite. Is the converse true? That is, given a Hausdorff topological space such that every compact subset is finite, ...
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20 views

Easy question about compactly contained sets

Let $U\subseteq X$ be a an open set of a topological space $X$ and $V\subset\subset U$ an open, compactly contained set (i.e, $\bar{V}$ is compact and $\bar{V}\subset U$). When we say the closure of ...
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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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Is every continuous one-to-one image of $[0,\infty)$ locally compact?

Suppose $f:[0,\infty)\to Y$ is continuous and one-to-one onto $Y$. You may assume $Y$ is metric. Is $Y$ locally compact? Thanks!
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Cantor Space - Example - Proving Compactness/Perfectness/Closed/Totally disconnected

Say we take the set of infinite binary codes $\{0,1\}^\mathbb{N}$, which is often written as $2^\mathbb{N}$, mapped to the Cantor set defined previously as $C_n=\frac{c_{n-1}}{3} \cup \left( ...
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Zero sets in Mrówka spaces

For a maximal almost disjoint family $\mathcal A$ of subsets of $\omega$ we choose a set $\{x_A:A\in\mathcal A\}$ of distinct points not in $\omega$ and define $\Psi (\mathcal A)=\omega\cup \{x_A:A\in ...
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A set A is compact, is its boundary compact?

I am trying to understand the concept of a boundary, and I have seen it defined $Bd(A) = \overline{A} \cap \overline{A^{\complement}}$. I was wondering three things, First how can I show that the ...
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1answer
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Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
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41 views

if $X$ has a finite number of isolated points, is $X$ compact?

If every real valued continuous function on $X$ is uniformly continuous is $X$ is compact? Moreover if $X$ has a finite number of isolated points, is $X$ compact now? I think that the answer to the ...
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1answer
37 views

$X $ is complete if every real valued continuous function on $X $ is uniformly continuous

If every real valued continuous function on $X $ is uniformly continuous,then is $X$ complete? My attempt:let $x_n$ be a Cauchy Sequence in $X$. Let $f$ be a real valued continuous function. To show ...
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1answer
38 views

show a subset of $\mathbb R^n$ is compact if it is closed and bounded

Use the two lemmas to prove that a subset of $\mathbb R^n$ is compact if it is closed and bounded. Lemma 1: A closed subset of a compact space is compact Lemma 2: If $X$ and $Y$ are compact then ...
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Compactness of two equivalent metric spaces

Let $X$ be a non-empty set. Suppose that $d_1$ and $d_2$ are two possibly different metrics on $X$. Let $\tau_i$ denote the topology generated by the metric $d_i$ ($i\in\{1,2\}$). The following are ...
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What does it mean for a set to be compact? (intuitively)

I'm having trouble intuitively understanding what it means for a set to be compact. I know that by definition a set is compact if for every open cover of the set there exists a finite subcover. But I ...
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88 views

How can I prove that something is an open cover?

I want to prove that the union of some intervals forms an open cover for some segment. Any ideas on how to do this?
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Would a connected space contain a compact subspace

I am trying to prove that in a connected space - $X$ , for every two elements of $X$ - say $a,b$ I can find a subspace of $X$ ( say $X'$ ) , such that$ X'$ contains a,b and is also connected, and ...
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$C_b(X)$ is non-separable for $X$ non-compact

If $X$ is a non-compact space then prove that $C_b(X)$ is not separable, where $C_b(X)$ is space of all bounded continuous functions on $X$. I was trying like this, but got stuck at middle: Take a ...
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33 views

Examples of non-compact connected spaces with the property…

I am looking for a non-compact connected space $X$ such that for any two disjoint closed $A,B\subseteq X$ there exists a proper closed connected $C\subseteq X$ such that $A\cup B\subseteq C$. I ...
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Proof with compact sets in Hausdorff space

Prove that every compact set in Hausdorff space is closed. Let $(X,\tau)$ be Hausdorff space and $A,B$ compact, disjont subsets of $(X,\tau)$. Prove that exist two disjoint sets $V,W$ open in ...
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Differentiable Manifold minus point not compact

Suppose $X$ is an $n$-dimensional for differentiable manifold for $n \geq 1$: in our definition this is a second countable Hausdorff space with a maximal differentiable atlas. If $p \in X$ is a ...
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Showing compactness of complete metric space

I need to show that for $K>0$, $$X=\{f:[0,1]\rightarrow [0,1]\mid |f(x)-f(y)|\leq K|x-y|\ \forall x,y \in [0,1]\}$$ with the metric $d(f,g)=\max|f(x)-g(x)|$ , (supremum metric), is a compact ...
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Give an example of a set that is closed but not compact nor bounded. Prove your answer.

Let $X = (0,\infty)$ with the usual topology in $\mathbb{R}$ and the the usual metric. Consider $A \subset X$ where $A = [1, \infty)$. Then $A$ is closed as $A' = (0,1) \subset X$. My attempt is as ...
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Closedness of Continuous Mappings from Compact Metric Space to Compact Metric Space

Let $(X, \rho_{X})$ and $(Y, \rho_{Y})$ be two compact metric spaces. Consider the metric space $(M_{XY}, \rho)$, where $M_{XY}$ is the set of any mappings from X to Y and $\rho(f,g) := \sup_{x \in ...
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prove: a complete metric space $X$ is compact if and only if …

Let $X$ be a complete metric space. Suppose that for any infinite subset $A$ of $X$ and for any $\epsilon>0$ there are $x_1,x_2 \in A$ such that $d(x_1,x_2)< \epsilon$. Show that $X$ is ...
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continuity of a map on a $T_2$ space

Let $X$ be a $T_2$ space .Let $f:X\rightarrow \mathbb R$ be such that $\{(x,f(x):x\in X\}$ is compact.Show that $f$ is continuous My attempt: Let $x_n$ be a sequence in $X$ converging to $x$.To ...
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Compact subspace of $\mathbb{R}$ with lower limit topology must be countable.

Any compact subset of $\mathbb{R}_{l} $ must be a countable set. Consider the open cover {[n,n+1): n is an integer} of R which has no subcover. So R is not compact with respect to lower limit (or ...
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If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact.

If $X\neq\varnothing$ and $\tau=\{\varnothing, X\}$, then any subset of $X$ is compact. Disproof by counterxample? Not true. Let $X = \mathbb{R}$ with the usual topology and $A = (-\infty,0)$. ...
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Problem of a compact space

Let $X$ be a compact $T_2$ space.Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional.Show that $X$ is finite. Spent nearly 3 hours on this problem.Cant ...
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Give an example of an infinite compact set $A$ such that its supremum is not a limit point

I got this one on a quiz the other day (We're only working in the reals). My solution was $$A=[0,1]\cup\{3\}$$ The closed interval has the infinite points, and $\sup A=3$ is not a limit-point since ...
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Proof with set compactness with river metric

We have got $d_r$ metric $$d_r(x,y) = \begin{cases} |x_2-y_2|, & \text{if $x_1 = y_1$;} \\ |x_2| + |y_2| + |x_1-y_1|, & \text{if $x_1 \neq y_1 $} \end{cases}$$ Prove that inside ...
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$A \subset \mathbb{R}^n$. If every continuous function $f: A \rightarrow \mathbb{R}$ is is bounded and attains its bounds then A is compact.

I'm doing a metric spaces course and got stuck on proposition. I have a feeling that I want to show that $A$ is bounded and closed then use Heine-Borel theorem. The proposition states that $f$ is ...
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1answer
52 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
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Principal ideal rings are not FO axiomatizable

A ring $R$ is a principal ideal ring if it is a ring and a model of $\forall I[I \text{ is an ideal} \to \exists x \forall y(y \in I \leftrightarrow \exists z (y = z*x))]$. How can one prove that this ...
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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 Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}.r I have to show that if A={a} is a singleton, then d(A,B)>0 and I have no idea how to do this.
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Show that the set of uniformly Lipschitz functions vanishing at $0$ is compact in $C[0,1]$

The question is: For $K$ and $\alpha$ fixed, show that $\{f\in \operatorname{Lip}_k \alpha : f(0) = 0\}$ is a compact subset of $C[0,1]$. I was going to attempt this by using by Arzela-Ascoli theorem ...
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Show $K$ is Compact.

I'm new to Real Analysis and didn't know how to start this question: Let $x_n \rightarrow x$ in $(M, d)$. Let $K = \{x\} \cup \{x_n : n\in\mathbb{N}\}$. Show that $K$ is compact. I wanted to try ...
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Motivation for the Definition of Compact Sets

I'm currently taking my first course in real analysis, and was recently introduced to the following definition of compact sets: A set $S \subseteq \mathbb{R}$ is compact if and only if every open ...
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compactness criterion for random variables in L2

Suppose $X_n$ is a sequence of random variables such that their second moments are uniformly bounded. I would like to know a compactness criterion for this case. In analysis, if $K$ is a bounded ...
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1answer
16 views

Compactness of the sum of line segments

Let $A\subset(0,\infty)$. Now $X(A)\subset\mathbb{R}^2$ will be a sum of closed intervals connecting points $(0,-1)$ and $(a,\frac{1}{a})$, $a\in A$. I am asked to prove $X(A)$ is compact $\iff A$ is ...
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Compact set in R that is not convex?

Just need an example. For example, the I know the set [0,1] is compact because it is obviously closed and bounded. But I have no idea how to test for convexity
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Would the following proof be wrong? (About the intersections of compact subsets)

Let $X$ be a topological space, and let $\{K_\alpha\}_{\alpha\in A}$ be a family of closed compact subsets of $X$. Show that $\bigcap_{\alpha\in A} K_\alpha$ is compact. Proof: Let $\mathcal{T}$ ...
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Kelley's topology : A topological space X is compact iff each nest of closed non-void sets has a non-void intersection.

Recall that a nest is a family of sets which is linearly ordered by inclusion. This problem is from kelley's "general topology" problem 5.H. the necessity follows from the finite intersection ...
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Show that the unit sphere with centre $0$ in $\mathbb{R}^d$ is compact.

Namely, the sphere is $\{x\in\mathbb{R}^d: \| x\|_2=1\}$. I am going about this by proving that the sphere is bounded and closed. I have proved that it is bounded and I can see that it must be closed ...