# Tagged Questions

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### The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
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### If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
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### Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
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### A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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### Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...
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### Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
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### Sequence of compact sets

Let $(X,d)$ be a metric space and consider an increasing sequence $A_n$ of its subsets such that $A = \bigcup_n A_n$ is compact. Can it happen that $A\setminus A_n$ is compact for all finite $n$?
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### Show that a map with some properties is closed

Let X be a topological space and Y hausdorff and local compact. Let $f:X \rightarrow Y$ be a continuous map such that $f^{-1}(K)$ is compact for all compact sets $K$. Show that $f$ is a closed map. ...
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### A generalization of the Arhangelskii Theorem [migrated]

Arhangeleskii's Theorem states the following For any Hausdorff topological space $X$, $$|X|\leq2^{\chi(X)L(X)}$$ where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of ...
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### If X is local compact, then it holds: A is closed $\iff$ $A\cap K$ is compact for all compact K [closed]

Prove: Show that for every local compact space X holds the following: A $\subseteq$ X is closed $\iff$ $A \cap K$ is compact, for all compact sets K. I use the following definition of local ...
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### Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
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### Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
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### In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
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### How to check that finite sets are dense in exp(X)?

How i can check if finite set $\bigcup F_{n}$ is dense in $exp(X)$, where $exp(X)$ is $$exp(X)= \{ A\in X ; A\not= \emptyset ; A \textit{ compact in } X\}$$ ($exp(X)$ is hyperspace, so it is set ...
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### Topological property: set-theoretically large subsets of an infinite space are not compact.

Let $X$ be an infinite topological space. Say that $X$ satisfies # if no subset of $X$ of cardinality $|X|$ is compact. So for instance it is clear that no (infinite) compact space satisfies # any ...
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### Stone-Cech compactification: clarification of definition

When defining Stone-Cech compactification we take a Tychonoff space $X$, the space $C_b(X)$ of bounded continuous real functions on $X$, define $I_f$ as closed limited intervals containing $f(X)$ for ...
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### Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
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### let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$. > Then show that $A$ is nowhere locally compact.

let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$. Then show that $A$ is nowhere locally compact.
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### A set is compact if and only if every continouos function is bounded on the set?? [duplicate]

I was asked to prove the following statement: let $K \subseteq R^n$. show that $K$ is compact (meaning closed and bounded) if and only if every continouos function is bounded on $K$. What I did: ...
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### Compactness of a set of partitions

The interval $[0,1]$ is partitioned to $n$ disjoint parts. Is the set of all possible partitions compact? There are several cases: A. All $n$ parts are connected intervals (possibly empty). In this ...
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### If X is a space in the order topology with lub. If A is closed, is A compact?

In J R Munkres section 27, there is a theorem that states that every closed interval(note not ray) in the order topology where $X$ is a set with lub property is compact. I'm wondering if $X$ is a ...
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### Compactness and existence of Pareto-efficient cake partitions

I am trying to understand a fundamental statement in the theory of cake-cutting. BACKGROUND: There is a certain "cake" $C$ (a subset of $R^n$). The cake is divided among two agents, 0 and 1. Each ...
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### Approximation by finite sets

I'm reading the book "Topology and Order" by L.Nachbin. In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes: [A]lthough these spaces may be infinite, they admit ...
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### Closed and Compactness on $\mathbb Q$ (Multiple Choice)

Please help me regarding the following question. Consider $\mathbb Q$ with usual metric (i.e $d(p,q)=|p-q|$).Then which of the following are true? $\{q\in\mathbb Q|2<q^2<3\}$ is closed ...
Prove or disprove: If $X$ is a paracompact space, then Alexandroff Duplicate $A(X)$ of $X$ is paracompact. Thanks for any help. ...
I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any \$u: ...