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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generalize the question every every intersection of nested sequence of compact non-empty sets is compact and non-empty

I'm aware how to prove that the intersection of nested sequence of compact non-empty sets is compact and non-empty. but I want to generalize this question to transfer the hypothesis of having nested ...
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2answers
50 views

$A$ and $B$ compact in a hausdorff space implies $A\cap B$ is compact [on hold]

Prove that if $A$ and $B$ are compact subset of a hausdorff space $X$, then $A$$\cap$$B$ is compact.
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2answers
60 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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1answer
10 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
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2answers
62 views

Quotient Maps and Compact Hausdroff Spaces

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Prove that if $X$ and $Y$ are compact Hausdroff space and $f:X\rightarrow Y$ is a continuous ...
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2answers
217 views

Are locally compact Hausdorff spaces with the homeomorphic one-point compactification necessarily homeomorphic themselves?

When practicing old qualifying exam problems, I had trouble with this one. Thanks for any help! Is it true that if the $1$-point compactifications of two locally compact Hausdorff spaces $X$, $Y$ are ...
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2answers
45 views

How to show that there exists a sequence in $[0,1]$ such that the set of accumulation points of the sequence is $[0,1]$

This is related to homework but I am trying to find a special case first and see if I can generalize it. The problem is to construct some sequence $(x_n)$ in $[0,1]$ such that the accumulation points ...
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1answer
27 views

Compactness and convergence

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
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1answer
18 views

Is there a lower bound for the maximal number of separated sets?

Let $(X,d)$ be a metric space and $T\colon X\to X$ uniformly continuous. A set $E\subset X$ is said to be $(n,\varepsilon)$-separated if for any distinct $x,y\in E$ there is a $0\leq j< n$ such ...
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2answers
25 views

Compactness and Hausdorffness with different topology

Here is the question (Munkres pg. 170): Show that if $X$ is compact Hausdorff under both $\mathcal{T}$ and $\mathcal{T}'$, then either $\mathcal{T}$ and $\mathcal{T}'$ are equal or they are not ...
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1answer
19 views

Limit of bounded functions in compact-open topology

Let $(X, \mathscr{T})$ be a topological space and $(Y,d)$ be a metric space. Recall that the compact-open topology $\mathscr{T}_{co}$ on $Y^X$ is generated by the subbase $$ \mathscr{S} = \{ S(C, V) ...
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1answer
58 views

When is the dual ball of $L_1(\mu)$ weak*-sequentially compact?

Where could I find a direct proof showing that the dual ball of $L_1(\mu)$ is weak*-sequentially compact? Since $(L_1(\mu))^*=L_\infty(\mu)$, I mean the unit ball $B_{L_\infty(\mu)}$ of ...
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1answer
46 views

Show For any language L two L-structures M and N are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage.

Setting For any language $\mathcal L$, two $\mathcal L$-structures $\mathcal M$ and $\mathcal N$ are elementarily equivalent iff they are elementarily equivalent for every finite sublanguage. ...
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1answer
25 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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1answer
27 views

Does $\sigma$ -compact imply separable?

Let $D$ be a metric space. If $D$ is $\sigma$-compact, does this imply that $D$ is separable? I thought I had a proof, but I think it is wrong. my proof: Let $K_n$ the compact sets such that $K_n ...
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2answers
353 views

Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
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2answers
85 views

In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Prove that if $X$ is Hausdorff and $\mathfrak{C}$ is a nonempty chain of compact and connected subsets of $X$, then $\bigcap \mathfrak{C}$ is compact and connected. Here are the definitions which ...
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0answers
40 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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0answers
40 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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1answer
42 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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2answers
158 views

A subset of a metric space is closed iff its intersection with every compact subset is closed

I want to show that a subset of a metric space $X$ is closed iff its intersection with every compact subset of $X$ is closed
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1answer
25 views

Proving subsets of $l^{\infty}$ are compact

Recently I started reading up on some set theory and metric spaces. I just read about compact subsets and I thought I understood it but in the exercises I'm having difficulty with the following ...
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1answer
27 views

Is compact $T_1$ topological space hausdorff?

I'm in a middle of a very hard exercise which its goal is to prove that some space is hausdorff, but all I could show is that it is $T_1$. But I can also deduce that it is compact. Is that enough for ...
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0answers
34 views

Non Satisfiability of disjuction

Problem: If S1,S2 are (possibly infinite) sets of propositional formulas where their union: S1VS2 is not satisfiable, prove that there exists an ψ such that S1|=ψ and S2|=¬ψ. Can we say that if ...
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1answer
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Is Alexandroff duplicate compact?

Consider the Alexandroff duplicate $X\times_{ad} 2$, the space $X\times 2$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X$, $(U\times\{0,1\})\setminus (x,1)$ is ...
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1answer
294 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 ...
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83 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
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1answer
17 views

Bounded set that is not closed nor compact

I am to find a set that is bounded but not closed nor compact. Here are my ideas. Please tell me if any of my logic is flawed. I thank you in advance. Consider the set $A = (0,1)$ where $A \subset ...
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1answer
345 views

$f:M_1\to M_2$ is continuous iff its graph is compact.

I have a propostion in Introduction to Real Analysis (3rd Ed.) which says: If $M_1$ is compact, a function $f:M_1\to M_2$ is continuous iff its graph is compact. Here $M_1$ and $M_2$ are ...
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1answer
51 views

Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
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2answers
24 views

Negate a proposition with quantifier?

I'm going over the proof of the theorem stating that "In a metric space, compactness impliess sequential compactness". I'm very likely confusing myself. I have the following proposition: $\forall ...
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0answers
34 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
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0answers
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Uniform best approximation in Chebyshev/Haar systems and the necessity of compactness of the function domain.

A great deal of Chebyshev/Haar systems are given for intervals $]-\infty,\infty[$, $[0,\infty[$ and other noncompact subsets of $\mathbb{R}$. Nonetheless, the theory of uniform best approximations in ...
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2answers
520 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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6answers
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Every compact metric space is complete

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
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2answers
704 views

Understanding Alexandroff compactification

Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
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2answers
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Prove that a compact metric space is complete.

I'm reading Intro to Topology by Mendelson. I'm in the section titled "Compact Metric Spaces". The problem is in the title. My attempt at the proof is as follows: Let $\{a_n\}_{n=1}^\infty$ be a ...
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1answer
14 views

Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let ...
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0answers
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Compactness result

I try to prove this lemma: Let $\mathrm{H}_{\text{comp}}^1(\Bbb R^{\mathrm{N}})$ be the subspace of $\mathrm{H}^1(\Bbb R^{\mathrm{N}})$ of functions with compact support. For each ...
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2answers
140 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 ...
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1answer
231 views

Properties if the one-point compactification of an uncountable discrete space

Let $D ( \tau )$ be an uncountable discrete space, and $\alpha D ( \tau )=D ( \tau )\cup\{\alpha\}$ the one-point compactification of $D ( \tau )$. I want to show that if $U$ is any countably ...
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Proving some property of a set of logical expressions that satisfies some properties

I am stuck at this problem. Let $\Sigma$ be a (finite/ infinite) set of logical expression (I.e. strings of the form $(P\land Q)$ or $\lnot(P\lor \lnot (Q\land R))$ etc.). That satisfies the ...
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1answer
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Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
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Compact Subsets [closed]

I drastically need help with these questions. I have been working on this last problem for hours and do not even know where to start or what I am doing. The questions are: a) Let $K$ be a compact ...
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1answer
186 views

Proving that a sequence in $L^2(\mathbb R)$ is relatively compact

I have a bounded sequence $\{f_n\}_n$ in $L^2(\mathbb R)$ such that $\mbox{supp } f_n$ is uniformly bounded and $$ \int_{\mathbb R} x^2 |\Theta_n(x) (F f_n)(x)|^2 dx \leq C^2 $$ for all $n$, where ...
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1answer
42 views

Set of all orthogonal matrices over $\mathbb C$ is compact/not

How to show the fact that the set of all orthogonal matrices over $\mathbb C$ is compact By an orthogonal matrix over $\mathbb C$ I mean a matrix $A$ satisfying $AA^T=I$ and here $A^T=(a_{ji})$ where ...
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2answers
34 views

why union and Cartesian product of infinitely many compact sets is not compact

I'm aware that the union and Cartesian product of finitely many compact sets is compact, but why we can't generalize it to the union and Cartesian product of infinitely many of them? for example for ...
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1answer
32 views

Show that $R$ is closed but not sequentially compact.

Show that $R$ is closed but not sequentially compact. Attempt: A subset E of a metric space X is said to be sequentially compact if and only if every sequence $x_n \in E$ has a convergent ...
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1answer
30 views

Showing a mapping is a Homeomorphism

I am trying to prove that the Stone Cech Compactification map is a homeomorphism. I have most the proof finished, but I am stuck on showing that the inverse function is continuous. Here is what I have ...
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2answers
370 views

Why is the inverse image of a compact set under a special sort of function compact?

Let $f$ be a continuous closed function from $X$ to $Y$ where $X$ and $Y$ are topological spaces. (Closed means that for any closed set $C$, $f(C)$ is also closed). Suppose that for any $y$ in $Y$, ...