4
votes
1answer
39 views

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 ...
2
votes
1answer
54 views

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 ...
1
vote
1answer
38 views

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 ...
1
vote
2answers
54 views

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.
1
vote
2answers
51 views

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 $n$?
2
votes
3answers
76 views

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. ...
5
votes
0answers
32 views

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 ...
2
votes
2answers
51 views

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 ...
1
vote
1answer
49 views

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 ...
1
vote
1answer
83 views

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...
0
votes
2answers
56 views

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 ...
1
vote
1answer
97 views

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 ...
4
votes
1answer
35 views

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 ...
3
votes
1answer
47 views

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 ...
3
votes
1answer
36 views

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 ...
-1
votes
1answer
29 views

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.
2
votes
1answer
63 views

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: ...
1
vote
1answer
55 views

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 ...
0
votes
0answers
40 views

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 ...
2
votes
1answer
53 views

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 ...
2
votes
1answer
37 views

In a compact space, every net has a convergent subnet

I'm just learning how to work with nets. I'm attempting the proof that $X$ compact $\implies$ every net in $X$ has a convergent subnet, and I wonder if I'm overcomplicating it. Suppose $\langle x_i ...
0
votes
0answers
53 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
0
votes
3answers
56 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
3
votes
1answer
57 views

Compact Hausdorff topological spaces

If we have a topological space $(X,\mathcal{T})$, where it is compact and Hausdorff, them we can say that any other topology $\mathcal{H}$ on $X$ such that $\mathcal{T}\subseteq\mathcal{H}$, the ...
4
votes
1answer
42 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
0
votes
0answers
21 views

Stone-Čech compactification not by ultrafilters only.

I am familiar with Stone-Čech compactification using ultrafilters. But, I, somehow can't understand the construction by commutative diagram, and certainly can not see the connection between the two ...
0
votes
1answer
37 views

For compact subspaces $C$ and $K$ of $X$ and $Y$, prove that for every open set $U$ of $X \times Y$, there exist open sets $V$ and $W$ with…

Let $C$ be a compact subspace of $X$ and let $K$ be a compact subspace of $Y$ . Let $U$ be an open set in $X \times Y$ containing $C \times K$. Show that there exist open subspaces $V$ of $X$ ...
0
votes
0answers
29 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
1
vote
1answer
33 views

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 ...
5
votes
2answers
44 views

Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
3
votes
2answers
33 views

How to “convert” from net to sequence in a first countable space

In a first countable space, what's a good way of going from nets to sequences? Let me explain more clearly what I mean. Suppose $f:X\to Y$ is a topological map and $X$ is first countable. Then I ...
8
votes
2answers
74 views

Topological distinguishibilty of $\infty$ after one point compactification?

Let $X$ be the one point compactification of some locally compact Hausdorff space. Let $\infty \in X$ represent the added point. Is there always a homomorphism $\phi:X \to X$ with $\phi: \infty ...
2
votes
1answer
61 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
1
vote
1answer
49 views

Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.

Say that a topological space is CH if it is both compact and Hausdorff. Let $T$ and $T'$ be two topologies on the same set X that are comparable but different, i.e., $T$ is either strictly ...
2
votes
1answer
43 views

I need a feebly compact topological space that is not pseudocompact

A Tychonoff topological space X is called pseudocompact if every continuous real-valued function with domain X is bounded. A space is called feebly compact if every locally finite family of open sets ...
0
votes
1answer
35 views

question about Stone Čech compactification

$x$ is normal space and we recognize him by his picture in $βX$. show that every $c_1 c_2$, close and disjoint sets in $x$ also the closure of $c_1$ and $c_2$ (in the closure of $x$) is disjoint. i ...
1
vote
1answer
35 views

Exapmles for Stone–Čech compactification

I'm finding it a bit though to "feel" this topic of Stone–Čech compactification. For example, I want to show that $[0,1]$ is not a Stone–Čech compactification of $(0,1]$ and on the other hand ...
1
vote
1answer
43 views

Distance between any two points in a compact metric space

I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ...
0
votes
2answers
34 views

X remains Subspace under Compactification

We just had the definition of compactification in our lecture, which says that Y must be compact and X must be a dense and open subspace of X. However in his Notes he gave the Definition so that X ...
0
votes
1answer
37 views

Relations between closed and compact sets

I have the following doubts: consider the set $\mathcal{Y}=[0,1]$ which is closed, convex, compact. Let $\mathcal{F}$ be the collection of closed subsets of $\mathcal{Y}$ and $\mathcal{K}$ be the ...
0
votes
0answers
35 views

How to prove that a sub-space of the functions $f: X \to Y$ is equicontinuous?

Let $X$ and $Y$ be two metric and compact spaces, and $C(X,Y)$ - the metric space of the continuous functions $f:X\rightarrow Y$. Denote by $Y^X$ the space of all functions (not just continuous) ...
0
votes
1answer
28 views

Do continuous functions preserve limit point compactness when the spaces are Hausdorff?

This is based on the problem in Munkres. Let $X$ be a limit point compact Hausdorff space and $Y$ a Hausdorff space. Let $f: X \to Y$ be continuous. Is $f(X)$ limit point compact in $Y$?
0
votes
1answer
17 views

Relatively compact sets in $\bar{\mathbb{R}}^{d}_{0}$

Reading an article I came across the following line, which botheres me since quite a while. Let $E:=\bar{\mathbb{R}}^{d}_{0}:=([-\infty,0)\cup(0,+\infty])^d$ be the closure of $\mathbb{R}^d$ without ...
1
vote
0answers
31 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
1
vote
3answers
50 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
-2
votes
1answer
54 views

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 ...
0
votes
1answer
45 views

Is Alexandroff Duplicate A(X) of X paracompact?

Prove or disprove: If $X$ is a paracompact space, then Alexandroff Duplicate $A(X)$ of $X$ is paracompact. Thanks for any help. ...
3
votes
0answers
89 views

Is every compact space compactly generated?

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: ...
0
votes
1answer
33 views

Compact and connected

Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by $$\{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...
1
vote
0answers
29 views

Topology - Compactness of $\mathbb{Z}\times\{0,1\}$

A question from my h.w.: Is the topological space $\mathbb{Z}\times\{0,1\}$ (where $\mathbb{Z}$ has the discrete topology and $\{0,1\}$ the trivial one) compact? sequentially compact? ...