# Questions tagged [compactness]

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. This includes logical compactness.

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### Compactness criterion for subsets in a fractional sobolev space

The compactness criterion of frechet kolmogorof gives necessary and sufficient conditions on when a set in $L^p$ is compact. Given a set of function in a Sobolev space $W^{k,p}$ one can apply that ...
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### Given any compact set in $\mathbb{R}^n$, the closure of the interior of the set is also compact.

In my Advanced Calculus class we had this excercise and I don't know if this proof is correct. Using the Heine-Borel Theorem, any set A in $\mathbb{R}^n$ is compact iff A it's closed and bounded. So ...
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### X/A with respect to the quotient topology is Hausdorff?

Consider $X = [0,1]$ and $A = (a,b)$ (where $0 < a < b < 1$). I want to prove $X/A$ is connected and compact. Here's my approach : If I take an open cover of $X/A$, the corresponding cover in ...
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### Prove that $\cup_{p \in M} \overline{B}(p, r)$ is compact

Let $M \subset \mathbb{R}^n$ be a compact subset of $\mathbb{R}^n$ and $r>0$. Prove that: $\bigcup_{p \in M} \overline{B}(p,r)$ Is compact. I know that this problem has an answer here: union of ...
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### Trying to compress information using functions. [closed]

This has been in my mind lately. Can we make an equation which can successfully compress a larger chunk of data using functions. For example, suppose we have a function $f(x)$ where $f(x)= 0$ or $1$, ...
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### Show that the unit sphere in the space of continuous functions $\mathcal{C}_p^0[0,1]$ is not compact

For $p \geq 1$, let $\mathcal{C}_{p}^0 [0,1]$ be the set of continuous functions in $[0,1]$ with the norm: $\lVert f \rVert_p = \left( \int_0^1 |f(x)|^p dx\right)^{\frac{1}{p}}$ Show that the unit ...
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### Where is the property "the intersection of compacts is compact" positioned among the lower separation axioms?

Every KC, i.e. "Kompacts are Closed", (and thus every $T_2$) space has the property I'll call IKK: the Intersection of any family of Kompact subsets is itself Kompact. Not all IKK spaces are ...
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### Munkres section 26 problem 2- Open coverings

Newbie in topology. Feels like I'm doing mental gymnastics and missing the point. I have the following two definitions and a lemma from munkres Which are helpful for problem 2(a) of Section 26 below ...
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### Compactness in angelic spaces

I have the following definitions: Definition 1: Let $X$ be a topological space and $A \subset X$. It is said that: $A$ is relatively compact if $\bar A$ is compact. $A$ is relatively sequentially ...
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### Proof verification for the Heine-Borel theorem

I am learning analysis by stopping at each theorem I encounter in the textbook and attempting a proof for it, until I am satisfied with the proof. For the Heine-Borel theorem however, although I have ...
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### Maximizing a convex function over an open convex set.

Let $f(x_1, ..., x_n)$ be a convex function. If I am maximizing the function over, say $[a_1, b_1]\times ... \times [a_n, b_n]$, then it is known that the maximum is attained at one of the extreme ...
1 vote
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### Bounded Variation In Compact set

Let $D$ be any finite collection interval on $E$ and function $f:E\to\mathbb{R}$. If $E$ is a compact set, show that V(f,E)=\sup\{V(f,D) \mid \text{for } D \text{ is any finite collection interval ...
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### Does a limit of a sequence exist in compact set

I am confused about the following statement in someone's paper: Since the probability simplex is compact, the sequence $\{a^{(T)}_n\}_{n\in [d]}$ belonging to the probability simplex for any $T$ ...
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### $C(X, \mathbb{C})$ separable [duplicate]

Let $(X, d)$ be a compact metric space. Is $C(X, \mathbb{C})$ separable? I've already checked an answer, but I think that my problem is that my continuous functions take value in $\mathbb{C}$. Does ...
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### Is the Borel sigma algebra compact w.r.t. Fréchet-Nikodym metric?

Let $\Omega \subseteq \mathbb{R}^d$ be bounded, consider the space $X:= \mathcal{B} (\Omega)$ equipped with the metric $d(A,B):=|A \triangle B|$ (where all sets with distance 0 are identified as usual,...
1 vote
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### If $Y$ is a metric space and $f: X \rightarrow Y$ is proper, $f$ is closed

Def: Let $f : X \rightarrow Y$ be a continuous map. $X,Y$ Topological spaces. $f$ is called proper if $f^{-1}(K)$ is compact for every compact $K \subseteq Y$. I want to prove that : If $Y$ is a ...
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### Metric space that can be written as the finite union of connected subsets but isn't locally connected

I'm looking for an example of a metric space $X$ such that for every $\epsilon > 0$ there exist connected subsets $A_1, \dots A_n$ for some $n \in \mathbb{N}$ such that $X = \cup_{i = 1}^nA_i$ and ...
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### Homeomorphism between Sphere and $[0,1]^2$

Following this question: Sphere homeomorphic to plane? I understand that a sphere is not homeomorphic to the plane because the sphere is compact and the plane is not. But why is the sphere not ...
1 vote
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### How to show that complete and pseudocompact implies sequentially compact (in a metric space)? [closed]

I have proved that pseudocompact implies completeness in a metric space, which is, as I understand it, a step to proving pseudocompactness implies compactness. How do I show complete and pseudocompact ...
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### Difficulty interpreting and solving Baby Rudin Problem 2.25, and how to optimally progress through Baby Rudin?

Context I am trying to solve Problem 2.25 in Baby Rudin. Here is the problem statement: (Rudin Problem 2.25)- Prove that every compact metric space has a countable base and that $K$ is therefore ...
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### If $\omega_f(y) \subseteq \omega_f(x)$ and $int(\omega_f(y)) \neq \varnothing$ then $\omega_f(y) = \omega_f(x)$

We let $X$ be a compact metric space and $f:X\rightarrow X$ a continuous function. For any point $x$ we define the orbit of $x$ under $f$ as $orb_f(x) = \{f^n(x): n \in \mathbb{N}\}$, where $f^n$ is ...
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### Is there anyone among us who can identify a certain SUS space?

The property US ("Unique Sequential limits") is a classic example of property implied by $T_2$ and implying $T_1$. In fact, it's the weakest assumption out of a chain of several distinct ...
1 vote
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### Decomposition of compact subset of a manifold

Let $K\subseteq M$ be a compact subset of a real $m$-dimensional manifold ($m\geq 1$). Let $\{U_i\}_i$ be an atlas of $M$. I am trying to show that then $K$ can be written as $K=K_1\cup\ldots \cup K_n$...
I let {${u_n}$}is a sequence of harmonic functions defined on an open disk and $|u_n|≤M$,where {$u_n$}satisfying $u_n(x)$→$u(x)$ for a.e.$x$ as n tends to infinity. And then I can't think of any way ...
$\mathbf {The \ Problem \ is}:$ Show that any open covering of a locally compact metrizable space $X$ can be refined to a canonical covering . We say that a closed set $F$ of a topological space $X$ ...