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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Alexandroff compactification [closed]
Let $X$ be a non-compact and Hausdorff topological space and $X^{*}=X\cup \{\infty\}$ be the Alexandroff extension of $X$ with topology on $X^{*}$, $\tau_{X^{*}}= \{U\subseteq X^{*}\mid U\cap X\in \...
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Inlcusion of compactly supported Sobolev Space in Closure of $C_0^{\infty}\left(\mathbb{R}^n\right)$ in $W^{k,q}(\Omega)$
Denote by $W_0^{k, q}$ the closure of $C_0^{\infty}\left(\mathbb{R}^n\right)$ in $W^{k,q}(\Omega)$ and by $W_c^{k, q}(\Omega)$ the set of functions from $W^{k, q}(\Omega)$ that have compact support in ...
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Proof that pointwise continuity implies uniform continuity for functions on compact sets
Theorem: Let $X$ be a compact metric space and $f : X→\mathbb{R}$ be such that $f$ is pointwise continuous for every $x \in X$. Then $f$ is uniformly continuous on $X$.
I was trying to prove this ...
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Rudin Real Analysis exercise 2.13, question about last part.
Self studying real analysis from baby rudin and stuck at this proof:
Trying to understand that proof for hours and stuck at the last part where the points of
$$ K \ \cap \ (x - \epsilon, x + \...
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The topology of a compact Hausdorff space $X$ is precisely the weak topology due to $C(X)$. How?
This question has been asked here, but the (upvoted) answer doesn't seem to be correct (see my comment below). Hence I re-ask the question:
If $X$ is a compact Hausdorff space, then is its topology ...
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Characterization by nets of relatively countable compact set on TVS
Let $E$ be a topological vector space and $A \subset E$. How can I prove that $A$ is relatively countable compact if, and only if, every sequence in $A$ has a subnet that converges in $\bar A$.
I know ...
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Compact and connected in quotient space
[In the question below, I use "compact" to mean "compact Hausdorff" and I use "quasi-compact" for what is commonly called "compact" in wikipedia and other ...
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Show that the set $C := \left\{\left.\sum_{j=1}^\infty\lambda_j e_j\right|\, \lambda_j \in \mathbb{K},\,|\lambda_j| \leq a_j\right\}$ is compact in H.
I am trying to work out question 8.17 from the book 'Functional Analysis: an elementary introduction' by Haase.
The question is formulated as follows.
Let $(e_j)_{j\geq 1}$ be an orthonormal system in ...
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Is the closure of $(0,+\infty) \cap (0, a)$ compact in $(0,+\infty)$?
Consider the half-line $(0,+\infty)$ as a topological subspace of $\mathbb{R}$. Suppose I consider an open interval $(0,a)$ for some $a > 0$. The set $(0,+\infty) \cap (0,a)$ is open in $(0,+\infty)...
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For $f:\Bbb R\to\Bbb R$ continuous, show that the family $A:=\{f(K)\,:\, K\text{ compact}\}$ is not a sigma algebra on $\Bbb R$
For $f:\Bbb{R}\to \Bbb R$ continuous, show that the family $A:=\{f(K)\,:\, K\text{ is a compact subset of }\mathbb R\}$ is not a sigma algebra on $\Bbb R$
I'm at loss here: how do I start? Which of ...
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Exercise 2 Chapter 4.1 - Magnus
I am trying to solve Exercise 2 (Section 4.1.6, Page 129) from Robert Magnus "Metric Spaces: A Companion to Analysis". I have tried to prove item (a), but I am a unsure on how to approach ...
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Intersection of closed and bounded subsets of $\mathbb{R}$
Let $\{ C_\lambda \}_{\lambda \in \Lambda}$ be a set of closed and bounded subsets of $\mathbb{R}$. Show that if $\bigcap_{\lambda \in \Lambda}C_\lambda = \varnothing$, then there is a finite subset $...
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Sequentially Compact implies Compact?
I know that in metric spaces sequentially compact is equivalent to compact. I also know that compact and sequentially compact for general topologies there is no relation between them. But I wanted to ...
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Integral of a continuous function continuous on a compact set w.r.t. the pointwise convergence topology
Take $K:[0,1]\times[0,1]\to [l,h] \subseteq \mathbb{R}$. For every $\tau\in [0,1]$, $K(\cdot,\tau)$ is continuous. Let $X$ be some compact subset of $[0,1]^{[0,1]}$. Does it follow that the map
$$
x \...
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homeomorphism between limit point compact set and 1st countable hausdorff space
X is a limit point compact space and Y is a 1st countable hausdorff space. Then show that bijective, continuous map f:X->Y is a homeomorphosm
All we need to show is f is an open or closed map. I've ...
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About compactness of simply ordered set with least upper bound property (Theorem 27.1 of Munkres)
I'm studying topology by Munkres.
Theorem 27.1:
Let $X$ be a simply ordered set having the least upper bound property.
In the order topology, each closed interval in $X$ is compact.
What makes me ...
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Confusion about why we need compactness
In my real analysis course, we need to prove that if $K\subseteq\mathbb{R}$ is compact, and $f:K\to\mathbb{R}$ is continous and injective, then $f^{-1}$ is continuous. The standard proof for this uses ...
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Compactness of the product of two sequences in negative Sobolev spaces
Let $K \subset \mathbb R^N$ be a compact set and let $f_n$ and $g_n$ be two sequences of functions from $\mathbb R$ to $\mathbb R$.
Suppose $f_n \to 0$ in $L^2(K)$ for $n \to +\infty$ and $\|g_n\|_{L^\...
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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 ...
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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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Compactness of bounded measures with respect to weak topology.
Consider the vector space of signed measures $\{m\}$ on $[0,1]$. Endow it with the weak topology generated by all functionals obtained as integral of a continuous function: $\int_{[0,1]} f dm, f \in C^...
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Locally invertible function is onto if its preimage on a compact subset is compact.
Question
Suppose $f:\mathbb R^n\to \mathbb R^n$ is a $C^1$ function and $Df(x)$ is invertible for all $x\in \mathbb R^n$. Then $f$ is onto if $f^{-1}(K)$ is compact for all compact set $K\subseteq \...
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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,...
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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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The Lebesgue measures of open sets of points of epsilon distance from a compact set converge to the Lebesgue measure of that compact set
I’ve found a few questions on this website relating to this question, but none that answer this specific question directly.
Let $E$ be a compact subset of $\mathbb{R}$ with Lebesgue measure $\lambda(E)...
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When does the Borel $\sigma$-algebra of compact convergence coincide with the product $\sigma$-algebra?
Let $X$, $Y$ be topological spaces, and $C(X,Y)$ the set of continuous functions $ X \to Y $, equipped with the compact-open topology. Let $\newcommand\Bco{\mathcal B_{\textrm{c-o}}} \Bco$ be the ...
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Boundless of $\nabla_{\theta} \log p_{\theta}$ with $p_{\theta}\in C^{1,1}$
let us assume to have a probability density function $p_{\theta}(x, y)$ defined on a compact set $\mathcal{X}\times \mathcal{Y}$ and with $\theta \in [0,1]^n$. Let us assume that $\theta\mapsto p_{\...
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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 ...
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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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62
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Is compact subset of a metric space a compact metric space? [closed]
Definition: Let $(X,d)$ be a metric space. We say $K \subseteq X$ is a compact subset of $X$ if every open cover for $K$ has a finite sub-cover.
Fact: $(K,d')$ can be regarded as a metric space where $...
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Compact subset inbetween another compact subset
Assume we have $A\subset \mathbb{R}^n$ open (regarding standard topology). If we have $B\subset A$ compact with dist$(B,\partial A)=\epsilon >0$, can we find $C\subset A$ compact with $B\subset C \...
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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 ...
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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 $K=K_1\cup\ldots \cup K_n$...
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Prove that if a sequence of harmonic function on the open disk converges uniformlyon compact subset of the disk，then the limit is harmonic.
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 ...
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Question regarding canonical covering of a locally compact metrizable space
$\mathbf {The \ Problem \ is}:$ 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 $X$ ...
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