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.

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$C$ is **weakly compact** or **weakly locally compact**?

Let $X$ be a separable Banach space such that $X$ and its dual $X^*$ have Radon-Nikodym property. Let $C$ be a convex, closed and bounded subset of $X$. Can we say that $C$ is weakly compact or ...
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How to prove this closure is compact?

Let $(X, d)$ be a compact metric space, and $\mathscr{F}$ an equicontinuous family of functions from $X$ to itself. Suppose that $g: X → R$ is continuous. Show that the family $\mathscr{G} = \{...
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$(X,\tau)$ topological vector space (Hausdorff) is: locally compact $\iff$ of finite dimension.

I am trying to show that in any finite dimensional normed vector space, the unit ball is compact. To do this, I first proved that: $(X,|\cdot |_X)$ normed vector space is: locally compact $\iff$ the ...
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Topological difference between the compact interval $I$ and the Cantor set

There is an homeomorphism between the Cantor set $X = 2^\omega$ (with the product topology) and the Cantor ternary set $\mathcal{C}=[0,1] \smallsetminus \bigcup_{n=0}^\infty \bigcup_{k=0}^{3^n-1} \...
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$ \frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact} $

Let $X$ be a separable Banach space. Let $C_1,...,C_n$ are nonempty weakly compact convex subsets of $X$. Why $$ \frac{1}{n}\sum_{i=1}^{n}{C_i}\text{ is weakly compact} $$ An idea please.
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Distinct topological compactifications

I'm looking for distinct topological compactifications (I know about the Stone-Cech compactification and the Alexandroff compactification). I would also like some examples of each (e.g. the ...
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42 views

On the definition of compactness

Rudin, in Principles of Mathematical Analysis, defines compactness: A set 𝐸 in a metric space 𝑋 is compact if and only if for any open cover $\{G_\alpha\}$ of $E$ there exist a finite subcover $G_{\...
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Compactification of log z Riemann Surface

I've been reading the 'Road to Reality' book of Roger Penrose and in the chapter on Riemann Surfaces, there is a note that we can compactify the log z Riemann Surface into a sphere. But I don't see ...
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Proof check of the fact that Compactness in Metric Spaces implies Closed

In apostol's analysis, this theorem is given a different proof(i feel so), but i wonder if this works as well(which is in fact the technique used to prove this theorem for the Euclidean Space $\mathbb ...
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Proving that a closed subset of $C[0,1]$ is compact.

Let $C=C[0,1]$ be the space of all continuous functions on $[0,1].$ $$K_n(a)=\{x.\in C:|x_0|\leq 2^n,|x_t-x_s|\leq N(a)|t-s|^a \enspace\forall |t-s|\leq 2^{-n}\},$$ where $N(a)=\frac{2^{2a+1}}{2^a-1}...
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Stone-Weierstrass Theorem with two compact metric spaces

I currently try to prove the following: Let $X,Y$ be compact metric spaces. $A = \{(x,y) \rightarrow \sum_{i=1}^{n} f_i(x)g_i(y) \ | \ f_i \in C(X,\mathbb{R})$ and $g_i \in C(Y,\mathbb{R}), 1 \...
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An exercise from Zorich II may be wrongly stated

The exercise can be found in the second volume, pag. 18: Show that a subset of a complete metric space is compact if and only if it is totally bounded and closed. I found this link: https://www....
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Upper semicontinuous decomposition

I'm reading a paper from Y. Ünlü called Lattices of compactifications of Tychonoff spaces. I've bumped into some definitions that I've never seen; while I find most of them understandable, there's one ...
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1answer
49 views

Does the property hold almost everywhere?

Suppose a property holds in every compact subset of $I$. Does it follow that this property holds almost everywhere on $I$? I want to use the dominated convergence theorem but do not know almost ...
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Real analytic set on a compact domain, no zeros on the boundary - isolated points only?

I have a feeling that the following must be true, but I cannot figure out a proof. I have two real analytic functions, $f$, $g$, both $[0,1]^2\rightarrow\mathbb{R}$. I am interested in the set for ...
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Intuitive way to think of the Heine-Borel Theorem in $\mathbb R^n$

When i encountered this theorem for the space $\mathbb R$, it was quite easy to come up with an intuitive understanding, i.e, thorugh the means of diagrams. But in the general space $\mathbb R^n$, its ...
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Let $X$ be compact and $f:X\to\mathbb{R}$ s.t. each $x\in X$ has a nbh where $f$ attains its minimum. Show $f$ attains minimum on $X$.

Consider a compact topological space $X$ and a map $f:X\to\mathbb{R}$ such that each $x\in X$ has a neighborhood where $f$ attains its minimum. Show that $f$ attains its minimum on $X$. My ...
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Product of a $k$-space and a compact space

I am beginning to learn about compactly generated spaces. I would like to know whether the following is true: if $X$ is a compact Hausdorff space and $Y$ is Hausdorff compactly generated space, then ...
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Brouwer's fixed point theorem in an infinite-dimensional space

I am wondering if the Brouwer's fixed point theorem can also be applied in an infinite-dimensional space. For example let $E = [0, 1] \times [0, 1] \times [0, 1] \times \dots$ be an infinite ...
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A subspace of a set of bounded and continuous functions is closed in this set and the ideal of this set

Question: Let $(X, d)$ be a compact metric space. For a given $x_0 \in X$, define $C_{x_0}(X,\mathbb{R})$ by $$C_{x_0}(X,\mathbb{R}) = \{f \in C(X, d):f(x_0) = 0\}$$ Note that $C(X,\mathbb{R})$ ...
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Image of a compact set through a continuous open mapping.

Suppose $X$ and $Y$ are metric spaces, $f:X\rightarrow Y$ is a continuous open mapping and $F\subseteq X$ is a compact set. Show for each $x\in F$ and $r>0$ there exists $w=w(r)>0$ such that $$B^...
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1answer
27 views

Locally compact subsests of R

I'm looking for an example of two locally compact subsets of the real line R, but their union isn't locally compact. I know that generally it is not true that such union is locally compact, as we ...
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Show that if $A$ and $B$ are compact subsets of $(\mathbb{R}^m,||.||_2)$ not empty and disjointed, then $\inf\{||a-b||_2:a\in A,b\in B\} > 0$

Show that if $A$ and $B$ are compact subsets of $(\mathbb{R}^m,||.||_2)$ not empty and disjointed, then $$\inf\{||a-b||_2:a\in A,b\in B\} > 0$$ I know the definitions and I been trying for a while ...
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'Completeness' of ordered topological space

This is a follow-up question to an answer given by Henno Brandsma in this thread How to prove ordered square is compact. In the answer it is shown that: A non-empty LOTS (linearly ordered ...
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32 views

An application of Uniform Boundedness Theorem

I did this problem before "Let $X$ be a compact Hausdorff space and assume that $C(X)$ is equipped with a norm $||.||$ with which this is a Banach space. For each $x\in X$, define $\lambda_x:C(X)\to \...
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How do we prove that compact spaces in metric spaces are bounded?

Let $(X,d)$ be a compact metric space. Then $(X,d)$ is both complete and bounded. My solution The space $(X,d)$ is indeed complete. This is because every Cauchy sequence which admits a convergent ...
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$K=\{f \in C^1([0,1]): f(0)=0, |f'(x)|\leq 1 \; \forall x\}$ can be covered with $4^n$ balls of radius $1/n$

I am trying to solve the following problem: Let $ K=\{f \in C^1([0,1], \mathbb{R}): f(0)=0, |f'(x)|\leq 1 \; \forall x\} \subset C([0,1],\mathbb{R})$ Prove that $K$ is precompact in $C([0,1])$ ...
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Criterion for compactness in a metric space

Of the following theorem (Zorich, Mathematical Analysis II, p. 18): I'm trying to understand the proof of the $ \Leftarrow $ implication. Before showing the proof in question, I post here a simple ...
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$\epsilon$-grid in compact metric spaces

From Zorich, Mathematical Analysis II, pag. 16: I believe that lemma 4 is wrongly stated, since if we were actually thinking of a metric space $(K, d)$ with $K$ compact (i.e. such that from any cover ...
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Generalization of Kolmogorov precompactness criterion

In the book Elements in functional analysis from Hirsch and Lacombe, the Kolmogorov precompactness criterion for families of $L^p$ functions is stated as follows: Theorem: Let $H \subseteq L^p(\...
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25 views

The direct limit of a sequence of compact normal spaces is regular

Suppose $X$ is a topological space equipped with the direct limit topology of the sequence $K_1\subset K_2\subset \cdots$ where each $K_n$ is compact Hausdorff. Thus a set $A\subset X$ is open [resp. ...
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Prob. 7 (c), Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a locally compact space under a perfect map is also a locally compact space

Here is Prob. 7 (c), Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is ...
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How to Motivate Open-Cover Formulation of Compactness in a Metric Space?

The open cover formulation of compactness always seemed to come out of nowhere for me. I've consulted many Analysis textbooks, but all of them have been like - 'Here's the open cover formulation, now ...
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Minimal number of balls in a cover of a compact set

Let $K \subseteq \mathbb R^n$ be compact. Let $r>0$. Can we cover $K$ by $N(r)$ balls of radius $r$, centered around points that belong to $K$, with $N(r) \le c \frac{1}{\text{Vol}(B(r))}$? ...
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1answer
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Prob. 7 (b), Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a regular space under a perfect map is also a regular space

Here is Prob. 7 (b), Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1} \big( \{ y \} \big)$ is ...
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Prob. 7 (a), Sec. 31, in Munkres' TOPOLOGY, 2nd ed: The image of a Hausdorff space under a perfect map is also a Hausdorff space

Here is Prob. 7 (a), Sec. 31, in the book Topology by James R. Munkres, 2nd edition: Let $p \colon X \rightarrow Y$ be a closed continuous surjective map such that $p^{-1}\big(\{ y \} \big)$ is ...
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Not every compact subset of $\mathbb{R}$ is the support of a continuous function

I saw a question on an analysis exam asking whether every compact subset of $\mathbb{R}$ is the support of a continuous real-valued function. The solution that was provided noted that a counterexample ...
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1answer
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Help with understanding of compact support on function with finite integral

If we have some function $f:\mathbb{R} \to[0,\infty)$ such that $\int_{\mathbb{R}} f(x)dx =N$ for some finite $N$, it intuitively makes sense that there exists some (compact) interval $S=[-M, M] \...
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Let $f:X\to Y$ be a map between first countable Hausdorff spaces s.t. $f^{-1}(K)$ is compact, for all compact $K\subset Y$. Show that $f$ is closed.

Let $f:X\to Y$ be a map between first countable Hausdorff spaces for which $f^{-1}(K)$ is compact, for all compact $K\subseteq Y$. Show that $f$ is a closed map. My attempt: Let $F\subseteq X$ ...
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A stricter condition of Dini's Theorem

I am trying to do a similar proof of Dini's Theorem but with a stricter condition. So, given $f: X \to Y$, where (X, d) and (Y, d') are metric spaces, and X is compact $f_n \to f$ pointwisely $d'(f(x)...
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Proving Compactness of a Set of Trapped Energies

I have a very simple question, but it requires a bit of background. Here it is: Let $p(x,\xi)=|\xi|_g+V(x),$ where $(x,\xi)\in \mathbb{R}^{2n},$ $V\in C_c^\infty(\mathbb{R}^n),$ and $g$ is a ...
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Is my proof of the Heine-Borel theorem correct?

Let $X$ be a subset of $\textbf{R}$. Then the following statements are equivalent (a) $X$ is closed and bounded (b) Given any sequence $(a_{n})_{n=0}^{\infty}$ of real numbers which takes values in $...
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Net compactness and relative compactness

I'm trying to understand the relation between the following conditions. I will assume that $X$ is a Hausdorff topological space and $A \subset X$. $\overline{A}$ is compact; Every net $\{x_{\lambda}\}...
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Homeomorphism from the coset space $G/G_x$ to the orbit $xG$

I'm currently working through a set of notes on fiber bundles and I'm struggling to prove something mentioned in my notes. Question Let $G$ be a topological group and $X$ be a Hausdorff, ...
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$T_1$ spaces where the closure of a compact set is not compact

I have many problems with this exercise: Give an example of $T_{1}$ Topological Space $(X,\tau)$ and a subset $Y\subset X$ compact such that $\bar{Y}$ is not compact. Now, honestly, I know this ...
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Image of limit point compact Hausdorff space is a compact?

Q: Prove that if $X$ is a limit point compact and Hausdorff space and $f$ is a continuous map sends $X$ to real numbers space, then the image of $f$ is compact subspace of $\mathbb{R}$? I know that ...
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Why this: relative topology of F is discrete for being F a countable subset, which contradicts its compacity (of F)?

Let's see $\Bbb R$ with the countable complement topology is not path connected. Let's suppose there is a function $f:[0, 1] \to \Bbb R$ which connects two points $x$ and $y$. If it was a path, then $...
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18 views

Integral zero does implies function zero on boundary?

Let $C \subset \mathbb{R}^m$ be a compact and convex set. Consider a function $f: C \rightarrow \mathbb{R}$ continuously differentiable on $C$ such that $$\int_C \, f(x) \, d\lambda(x) \, = \, 0 $$ ...
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49 views

Theorem 29.1 in Munkres's Topology

Could someone please explain the highlighted sentence in this proof? I understand that $C$ is contained in $X$, but I don't understand why that implies it is a compact subspace of $X$ given that it is ...
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46 views

Importance of this result about compactness

Consider this result: Let $(X,d)$ be a metric space, $E\subset X$ and $p\in E$. Prove that if $\,\forall r>0$ the set $\{x\in E : x\not\in B_r(p)\}$ is finite, then $E$ is compact I am not ...

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