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.

learn more… | top users | synonyms

1
vote
2answers
30 views

A locally compact Hausdorff space is compactly generated

I am having trouble showing one direction of the proof that a locally compact Hausdorff space is compactly generated. Specifically, my question is how do I show that: if X is a locally compact ...
4
votes
1answer
91 views

Compact subset in colimit of spaces

I found at the beginning of tom Dieck's Book the following (non proved) result Suppose $X$ is the colimit of the sequence $$ X_1 \subset X_2 \subset X_3 \subset \cdots $$ Suppose points in $X_i$ ...
9
votes
2answers
160 views

Why does this proof fail?

I'm reading some notes on topology, and the notes' author is trying to raise motivation to consider compactness by providing a theorem whose proof is built intentionally wrong, but I don't agree with ...
1
vote
2answers
67 views

Which of the Following Sets are compact (C.S.I.R 2015)

$\{ (x,y,z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 1 \}$ in the Euclidean Topology $\{ (z_1,z_2,z_3) \in \mathbb C^3 : z_1^2 + z_2^2 + z_3^2 = 1 \}$ in the Euclidean Topology. $\prod_{n=1}^{\infty} A_n$ ...
2
votes
3answers
93 views

Importance of Locally Compact Hausdorff Spaces

I mostly deal with measure and probability theory and quite often, whenever I look up something on wikipedia, I see the mathematical objects defined on a locally compact Hausdorff space. I have very ...
1
vote
1answer
74 views

compactness of sets in euclidean topology and product topology

Which sets are compact in euclidean topology and product topology ? $\{(z_1,z_2,z_3):z_1^2+z_2^2+z_3^2=1)\}$ in the euclidean topology. $\{z\in \mathbb C:|re(z)|\leq a\}$ in the euclidean topology ...
0
votes
1answer
53 views

on existence of supremum/infimum

It is known that minimum or maximum of a function does not always exist but the supremum/infimum usually tends to exist. Example 1: For example, if we consider $X$ as the set or rational numbers with ...
1
vote
1answer
43 views

Intermediate Value Theorem on $\mathbb{R^n}$

Let $S^2$ denotes the subset of $\mathbb{R^3}$ which includes the points $(x,y,z)$ s.t $x^2+y^2+z^2=1$ i.e the boundary of a unit sphere. Let $f$ be a continuous function from $S^2$ to $\mathbb{R}$ ...
0
votes
1answer
46 views

Compact subset of a non compact topological space

Define a topological space X that is not compact and define a set A ⊂ X that is compact. Use the definition of finite open subcovers to show that A is compact. Ok so I think that a topological space ...
7
votes
2answers
123 views

Intuition for Kuratowski-Mrowka characterization of compactness

Fact. A space $X$ is compact iff for every space $Y$, the projection $X\times Y\rightarrow Y$ is a closed map. The finite subcover definition of compactness seems reasonably intuitive: finite covers ...
2
votes
3answers
54 views

Continuity of improper integral with a continuous integrand.

I am a newbie in analysis and am trying to wrap my head around some continuity/compactness/finiteness concepts. Let $f(x,y):\mathbb{R}^2\mapsto\mathbb{R}$ be a continuous function in both $x$ and $y$ ...
4
votes
2answers
54 views

$X$ compact Hausdorff with $X=X_1\cup X_2$. If $X_1,X_2$ are closed and metrizable, show that $X$ is metrizable.

This is Exercise 9 from Section 34 of Munkres - Topology. Following the hint given, I've done the following:Since $X$ is compact, $X_1,X_2$ are compact metrizable and hence have countable bases. Let ...
0
votes
1answer
45 views

Compact metric space characterization (continuous real functions)

Prove that a metric space is compact iff every continuous real function on it is bounded. $\ f: X \mapsto Y$; $f[X]=A$; If X is compact, then we can find a sequence in X, $x \mapsto a$. Because f ...
4
votes
1answer
42 views

Tychonoff's theorem for products of finite discrete topologies?

I need the following specific version of Tychonoff's theorem: Suppose $\{X_\alpha\}_\alpha$ is a collection of finite sets endowed with discrete topologies, then $\prod_\alpha X_\alpha$ is ...
2
votes
1answer
29 views

Finitely addivite finite regular set function on real Borel $\sigma$ algebra is already a measure.

Let $\mu$ be a finitely additive set function on the Borel $\sigma$-algebra $B_{\mathbb{R}}$ with $\mu(\mathbb{R})<\infty$ and $\mu (A) = \sup\{\mu(K) \mid K \subseteq A, \ K \text{ compact} \} $ ...
1
vote
1answer
41 views

Uniform boundedness implies equicontinuity on compact domains?

Suppose $F:\mathbb R^2 \to \mathbb R$ is continuous. Assume $f_n$ is a uniformly bounded sequence of real-valued functions on $[0,1]$ such that for each $n, f_n'(x) = F(x,f_n(x)),x\in [0,1].$ Is ...
2
votes
1answer
38 views

Regularity of Borel measures on compact metric spaces

Let $(X,d)$ be a compact metric space and $\mu$ be a finite measure on the Borel $\sigma$-algebra $B_X$ on $X$. Then we have for all $A \in B_X$: $\mu (A) = \inf \{\mu(Q) \ \vert \ A\subseteq Q, Q \ ...
7
votes
1answer
95 views

Which spaces can be used as “test spaces” for the Stone-Čech compactification?

Stone-Čech compactification $\beta X$ of a completely regular space $X$ is defined by the following property: Let $X$ be a completely regular space. Let $i \colon X \hookrightarrow \beta X$ be an ...
1
vote
2answers
26 views

Sequential compactness implies boundedness

Suppose a set $K\subset \mathbb{R}$. Every sequence in K has a convergent subsequence that converges to a limit in K. Prove that K is bounded. I want to use contradiction. Suppose K is not bounded, ...
2
votes
1answer
44 views

Approximation property for Banach space and $l^{p}$

Let's consider a compact operator $T: X \rightarrow l^{p}, 1 \leq p < \infty$. I would like to check, whether it's possible to approximate $T$ by the operators of a finite rank with respect to an ...
2
votes
1answer
21 views

Compactification of non-locally compact space

Is there a space $X$ that is not locally compact, and a compactification $Y\neq \beta X$ of $X$, such that $\text{cl}_Y Z_1 \cap \text{cl}_Y Z_2$ is finite whenever $Z_1$ and $Z_2$ are disjoint zero ...
1
vote
0answers
26 views

How to use Farkas' lemma?

How can I prove, that the set $$P = \{(x, y) \in \mathbb{R}^{n+m} : Ax + By \geq c, \: x \geq 0^n, \: y\geq 0^m \}, $$ where $B \in \mathbb{R}^{m \times m} \;$ is positive semidefinite matrix, $A ...
3
votes
0answers
35 views

Rellich-Kondrachov compacteness theorem for the Euclidean space with Gaussian measure

Let $\gamma_n: \mathbb{R}^n\to\mathbb{R}$ be the Gaussian distribution function defined by $$ \gamma_n(x):=(2 \pi)^{-\frac{n}{2}} e^{-\frac{|x|^2}{2}}. $$ Let $d\gamma_n$ denote the following measure ...
10
votes
2answers
149 views

Is there a direct proof that a compact unit ball implies automatic continuity?

One of the fundamental theorems in functional analysis is that if $X$ is a Banach space (say over $\Bbb C$) with a compact closed unit ball, then $X$ is finitely dimensional. The usual proof is by ...
10
votes
1answer
54 views

Compact Metric Spaces and Separability of $C(X,\mathbb{R})$

Let $(X,d)$ be a compact metric space. Show that $C(X,\mathbb{R})$ is a separable metric space (space of continuous functions from $X$ to $\mathbb{R}$). I first showed that if $(X,d)$ is compact, ...
3
votes
1answer
45 views

Compact Metric Spaces which are Groups and Topologies

Let $(G,d)$ be a compact metric space which as well is a group. Assume that $(x,y) \mapsto xy$ is continuous as a map $G \times G \to G$ and that group inversion $x \mapsto x^{-1}$ is continuous as a ...
2
votes
1answer
57 views

Compact sets in the lower topology on $\mathbb{R}$ have a minimum

Let $S$ be a subset of $\mathbb{R}$ be nonempty. Show that in the lower topology, $S$ is compact iff $S$ has minimum. Note that the lower topology is not the lower limit topology. I tried to ...
2
votes
1answer
29 views

“Powers” of compact Hausdorff spaces

Suppose $X$ is a compact Hausdorff space and $\mathbf 2$ is the space whose underlying set of points is $\{0,1\}$ equipped with the discrete topology. Clearly $\mathbf 2$ is compact Hausdorff. Denote ...
2
votes
1answer
35 views

Real analysis: compact sets and intervals

I am wondering whether the following statement is true or not. Let $S\subseteq \mathbb{R}$ be a compact set such that $\{0,1\}\subseteq S$ and $S\subseteq [0,1]$. If $S\neq [0,1]$ then there exist ...
6
votes
1answer
36 views

Weirdly defined ball compact in $C^1([0, 1])$

Consider$$B := \left\{u \in C^2([0, 1]) : \sum_{i=0}^2 \sup_{x \in [0, 1]} \left|u^{(i)}(x)\right| \le 1\right\}$$as a subset of $C^1([0, 1])$. How do I see that it is compact in $C^1([0, 1])$?
1
vote
1answer
46 views

Examples compact sets

At the moment I try to understand the topic "proving compact sets". 2 examples: I want to ask, if my assumptions/conclusions are right. Example 1: $(x_1-1)^3 + x_2 \le 0\:,\:x_2\ge0$ This set is ...
1
vote
0answers
18 views

Kuratowski measure of non-compactness of unit ball and unit sphere.

Let X be any metric space. Let $\mathcal{M}_X$ denote the class of all bounded subsets of a metric space $X$. Definition: Let $(X,d)$ be a complete metric space. The function ...
1
vote
2answers
49 views

Compact connected space is the union of two disjoint connected sets.

This was mentioned in an article, but I have not been able to find a proof anywhere. Assume the original space has at least 2 points. The disjoint connected sets should be nontrivial (each has at ...
0
votes
1answer
59 views

Topology: Difference between Bolzano-Weierstrass Property and Sequential Compactness?

In a general topological space are these properties equivalent ? If not, is there a property (e.g. first countability) that metric spaces possess which makes them equivalent there ? Here are the ...
2
votes
0answers
118 views

Compactness in $\mathbb{R}^\infty$. (Compact ellipsoid)

Need some help with this question in my Real Analysis course. Take positive decreasing numbers $l_1 \ge l_2 \ge \cdots\ge l_n \rightarrow l_\infty \ge0. $ It is to be shown that the ellipsoid ...
1
vote
0answers
43 views

T/F: The set: $\{(x,y): \sin(x^{2012} +y^3) + x^2 + y^4 \le 1\}$ is a compact set in $\mathbb{R}^2$.

T/F: The set: $\{(x,y): \sin(x^{2012} +y^3) + x^2 + y^4 \le 1\}$ is a compact set in $\mathbb{R}^2$. I think it is false since I don't believe the set is closed nor bounded? Is that correct?
1
vote
1answer
35 views

Compact space - definition

I have a doubt at the definition of compact spaces. So if you have a topological space $X$, then $X$ is compact if every open cover of $X$ has a finite subcover. In other words, if $X$ is the union of ...
0
votes
1answer
42 views

One point compactification, Hausdorff space in which every point has a compact neighborhood. Show $X'=X\cup\{\infty\}$ is compact and connected

Let $X$ be a non-compact connected Hausdorff space in which every point has a compact neighborhood. Show $X'=X\cup\{\infty\}$ is compact and connected, $X'$ takes on the one point compactification, ...
8
votes
2answers
108 views

Let $X$ locally compact, Hausdorff and non compact. Ends.

Let $X$ be locally compact, Hausdorff and non compact. Prove that if $X$ has one “end”, then $X^\wedge - X$ , (where $X^\wedge$ is any Hausdorff compactification), is a continuum (=compact, ...
0
votes
1answer
35 views

Proving a set is compact!

This is the last one i need help with and the help is much appreciate as I seem to have found myself stuck and pretty much turned in a blank worksheet to my professor. He says these types of problems ...
0
votes
2answers
29 views

Compactness on set with two topologies

Question : Let $\Omega_1 \subset \Omega_2$ be two topological structures in $X$. Does the compactness of $(X, \Omega_2)$ imply that of $(X, \Omega_1)$? And vice versa? My attempt : Assuming the ...
1
vote
2answers
46 views

How to prove that $A \cap B$ is compact using the definition of compact set

Suppose that $A$ and $B$ are compact set in $\mathbb R^n$. Then $A \cap B$ is compact. Using Heine-Borel Theorem, since $A$ and $B$ are both closed and bounded, $A \cap B$ is also closed and ...
0
votes
1answer
42 views

Show a continuous function on a closed bounded interval is Lipschitz under the maximum (infinity) norm

I'm currently working on the following: Define the function $\psi: C[a,b] \to \mathbb{R}$ by $\begin{equation*} \psi(f)=\int_{a}^{b}f(x)\,dx \end{equation*}$ for each $f \in C[a,b]$. Show that ...
0
votes
1answer
46 views

Finding an open cover without finite sub cover

I am looking for an open cover of [1,2) that has no finite subcover. I'm thinking (1/n, 2-1/n). Does this work? I think it is certainly an open cover of [1,2), but i'm not sure if it has finite ...
2
votes
1answer
59 views

For uniformly continuous $f:X\to Y$, where $X$ is a totally bounded metric space, prove that $f(X)$ is totally bounded

I am faced with the following problem: Let $X$ be a totally bounded metric space. If $f$ is a uniformly continuous mapping from $X$ to a metric space $Y$, show that $f(X)$ is totally bounded. Is ...
0
votes
1answer
22 views

Equivalence of uniform convergence in metric spaces!

Let $f, f_1, f_2, \dots, f_n, \dots$ be continuous applications $f_i: M \rightarrow N$, $ f: M \rightarrow N$. Then, the following affirmations are equivalent: $(1)$ If $x_n \rightarrow x$ in M, then ...
0
votes
1answer
23 views

$x_n \rightarrow x \implies \lim_{n \rightarrow +\infty}$ IFF $f_n \rightarrow f$ uniformly in each $K \subset M$ compact

Let $f, f_1, f_2, \dots$ be continuous maps from M to N, M and N metric spaces. The, the following affirmations are equivalent: $(a)$$ x_n \rightarrow x \implies \lim_{n \rightarrow +\infty} f_n(x_n) ...
2
votes
2answers
46 views

How to show whether the following sets are compact or not:

How to show whether the following sets are compact or not: $1.\{(x,y,z)\in \mathbb R^3:x^2+2y^2-3z^2=1\}$ $2.\{(x,y,z)\in \mathbb R^3:|x|+2|y|+3|z|\leq 1\}$ I know in order for a set to be compact ...
0
votes
1answer
20 views

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point [duplicate]

M compact. $(x_n)$ converges if and only if $(x_n)$ has only one closure point. Show, with one example, that the compacity is necessary. My attempt: 1) Of course if $(x_n)$ converges, $(x_n)$ has ...
1
vote
1answer
47 views

Which spaces are relatively compact and connected?

Let $X=\{(x,y):x^2+y^2<5\}$ and $K=\{(x,y) :1\leq x^2+y^2\leq 2\text{ or }3\leq x^2+y^2\leq 4\}$. Then which are true: $1$.$X\setminus K$ has $3$ connected components. $2$.$X\setminus K$ has no ...