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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107 views

An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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122 views

Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?

The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
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47 views

Compactness of a set of bounded functions in the uniform norm

Let $T$ be a non-degenerate compact interval in $\mathbb R$ and $f:\mathbb R^2\to\mathbb R$ a strictly concave function such that (a) $f(0,0)=0$, (b) $f$ strictly increases in the first argument, and ...
4
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125 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
4
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119 views

Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
4
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77 views

Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
4
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283 views

If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?

Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map. Using the embedding $Y\hookrightarrow\beta ...
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58 views

A question on countably compact space under CH

The question is also posted here. A regular space $X$ is star compact (which implies pseudocompact) with $G_\delta$-diagonal star countable first countable $e(X)\le \aleph_0$ ( in fact it implies ...
4
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130 views

Constructing the support of a Borel measure

From Rudin, Real and Complex Analysis, Chapter 8, Problem 7, 1st Edition. Suppose $E$ is a compact set in $\mathbb{R}^{k}$ without isolated points. Show that $E$ is the support of a continuous ...
4
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62 views

Classes of compact spaces

Given a class $\mathcal{C}$ of compact Hausdorff spaces which is closed under countable products and continuous images. Let $\kappa>\omega$ be a cardinal number. Consider the class ...
3
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58 views

Compact family of Lip functions under the sup norm metric, proof verification.

Hi everyone I'd like to know if the following is correct, I'd appreciate your opinion and also any suggestion to improve my argument. Thanks in advance for your time. If $(K,d)$ is a compact ...
3
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56 views

Compactness and Lipschitz functions

I am very stumped by this question: Suppose (K, d) is a compact metric space. Let f be any function, f: K $\rightarrow \mathbb{C}$, not necessarily continuous. Prove that for any $\epsilon > ...
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94 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: ...
3
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70 views

US does not imply AB

We say that a topological space $X$ is: $AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...
3
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72 views

References for the Čech-Stone compactification of Hyper-Reals?

It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when ...
3
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59 views

Lie groups with structure constant $f_{abc} \neq f_{bca}$.

The structure constant $f_{abc}$ of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically $f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ...
3
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153 views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
3
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102 views

Prokhorov theorem in locally compact Hausdorff space?

Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ...
3
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95 views

Smash product of compact spaces

In the topology book I'm reading I found the following statement: The "smash product" (of two pointed spaces) is defined as $X \bigwedge Y=X \times Y/(X \times \lbrace*\rbrace \bigcup Y \times ...
3
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39 views

Strongly additive open cover

Call an open cover $\mathscr{U}$ of a metric space $M$ strongly additive if whenever $U,V\in\mathscr{U}$ and $U\cap V\ne\emptyset$, then $U\cup V\in\mathscr{U}$. Prove that $M$ is compact and ...
3
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58 views

A question on pseudocompact space

The exercise is from the handbook of set-theoretic topology page 161: Assume $\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact pseudocompact space ...
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46 views

Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
3
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84 views

What is the point of compactness and how does it look visually?

I just learned what the term compact means, as in compact sets. I know what the definition of compactness is, but I don't get what the real significance of it is. How should I try to think of it in a ...
3
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269 views

Sequential compactness vs. countable compactness

Firstly, I'll give the definitions of sequential compactness and countable compactness. Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent ...
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31 views

Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
2
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81 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
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92 views

$\ f \colon X \to X $ ,continuous function where X is compact,Hausdorff space.Show $\exists A$ st $f(A) =A$.

Suppose $\ f \colon X \to X $ is a continuous function from a compact,Hausdorff space to itself. Prove that there exists a subspace $A$ such that $f(A) =A$. I came up with an answer based on nets ...
2
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83 views

Measures with bounded total variation norm compact in $M(X)$?

Let $X$ be a separable, metric, compact space. (e.g. an interval in $\mathbb{R}$ like $[0,10]$). Let $M(X)$ be the set of all finite signed measures over $X$ with weak-*-topology (in probability ...
2
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105 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
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34 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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26 views

Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
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33 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
2
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95 views

Is this proof correct about compact sets inside open sets?

I've been solving the following problem: "If $U\subset\mathbb{R}^n$ is open and $C\subset U$ is compact, show that there is a compact set $D$ such that $C\subset \operatorname{int}(D)$ and $D\subset ...
2
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182 views

Every Borel set is the union of an increasing sequence of Bounded Borel sets?

I am currently working with the book by Halmos, and i can't quite get past this one. It states that: "Every Borel set can be written as an increasing sequence of Bounded Borel sets" In this case $X$ ...
2
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125 views

For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$

The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
2
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0answers
63 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
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0answers
148 views

Connectedness of the complement of a compact “small” subset of $\mathbb R^n$

Let $C$ be a compact subset of $\mathbb R^n$ and suppose that for every $\varepsilon >0$ there exists a finite family of open disks $B_i$ s.t. $C \subset \bigcup_{i} B_i$ and $\sum_i r_i \le ...
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38 views

Weakly compact operator on $c_0$ is compact

Show that if $T\in {\cal B}(c_0)$ and $T$ is weakly compact, then $T$ is compact. My attempt: $T$ is weakly compact, so there is a reflexive space $X$ , and operators $A\in {\cal B}(X,c_0) $ and $B ...
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27 views

Convex sets and convex polytopes?

Consider the set $\mathcal{X} \subset \mathbb{R}^d$ convex and compact. Which is the difference between the collection of compact convex subsets of $\mathcal{X}$ and the collection of convex polytopes ...
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34 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 ...
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0answers
35 views

Example of Stone-Weierstrass Theorem on a non-interval? (3-dim compact set for example)

So I'm familiar with the Stone-Weierstrass Theorem for closed intervals [a, b] but am now looking to prove it for a more general 3-dimensional compact set. That is, if f is continuous on the set X, f ...
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42 views

If an upper semicontinuous multivalued map is compact on a set, is it compact on the boundary as well?

I have stumbled upon the following problem during my research: Let $X$ and $Y$ be Banach spaces, $K\subset X$ nonempty, $F:\overline{K}\rightarrow 2^{Y}$ an upper semicontinuous multivalued map with ...
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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? ...
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25 views

Compactification via embeddings and extending continuous functions

My question comes from reading Munkres' Topology, the section on Stone-Čech compactification. To find the compactification $\mathrm{Y}$ of $\mathrm{X}$, we find an embedding h, $\mathrm{h}: X ...
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55 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
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53 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
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33 views

Show a subset A is compact if and only if the image of the map T(A) is compact

The question is as follows: Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = ...
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31 views

Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
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29 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
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42 views

$\sigma$-$\sigma$-compactness is $\sigma$-compactness?

I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact? I'm not sure if a countable union of countable unions is still a countable ...