# Tagged Questions

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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### 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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### 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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### 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 ...
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### 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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### $\overline{\mathrm{conv}} \{x_i : i \in \mathbf{N} \}$ 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 $y$ ...
### $X$ 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 ...