All Questions
77 questions
0
votes
1
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44
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Ahlfors' complex analysis proof doubt - Heine-Borel theorem
I am currently reading Ahlfors' complex analysis. I am confused by his proof that a complete, totally bounded metric space $S$ satisfies the Heine-Borel property (by Heine-Borel property we mean any ...
- 29
3
votes
0
answers
34
views
Questions about S. Solecki, Analytic ideals and their applications.
I read from "S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999),51–71."
In the first; I can't understand what is the analytic ideal? I know what the ideal ...
- 373
1
vote
0
answers
32
views
On the intuition of compactness in terms of the derived set
The usual definition of a compact set $K$ is that from every open covering of $K$, there exist a finite subcovering. This definition gives the idea that the notion of compactness is kind of a ...
- 645
0
votes
2
answers
160
views
Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Let $(X,d)$ a metric space. Prove that if $K \subseteq X$ is compact and $F \subseteq X$ is closed, then $K \cap F$ is compact
Here's my proof so far, please check it.
Proof.
Assume that $K \subseteq ...
- 17
0
votes
1
answer
76
views
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,...
- 21
1
vote
0
answers
55
views
Baby Rudin 2.40: can an infinitely small subcover $G_\alpha$ be backward-constructed?
My question is about exactly how small can a nbhd of $x^\ast$ be, or put in another way: is it possible that some subcover $G_\alpha$ be constructed along the way so that it each time (albeit narrowly)...
- 129
0
votes
0
answers
61
views
Is C(R) compact space in regard to given metric
Let $C(\mathbb{R})$ be a set of continous functions $f:\mathbb{R}\to\mathbb{R}$ and $d:C(\mathbb{R})\times C(\mathbb{R})\to \mathbb{R}$ defined as
$$d(f,g)=\begin{cases} 0 , & f=g ;\\
\sup\limits_{...
- 6,615
7
votes
0
answers
110
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(Short proof verification) Problem related to distance to a set
I have this previous result:
Let $(E, d)$ be a metric space and $K\subset E$ a compact subset. Then, for all $x\in E$ there is a $k_x \in K$ such that $d(x, K)=d(x, k_x)$, where $d(x, K)=\inf\{d(x, k):...
- 1,364
0
votes
2
answers
423
views
Determine the open cover for set
The set is given as subset $X = \{(x, y) | x > 0, y > 0\}$ of $\mathbb{R}^2$. The set is open. Find an open cover for the set that does not admit a finite subcover. The purpose is to show that ...
- 503
0
votes
0
answers
41
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Concepts related to compactness
I recently took a second course in analysis and learnt about the definition of compact spaces. I was wondering if there is any significance to the following (even stronger) properties that a space $X$ ...
- 2,495
0
votes
1
answer
62
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How to prove this intersection of compact sets doesn't have the finite intersection property?
If $\{A_i\}_{i \in \mathbb{N}}$ is a countable family of compact subsets of a metric space $(X, \rho)$ such that $\bigcap_{i \in \mathbb{N}} A_i = \emptyset$ then there exists a finite subset $I \...
0
votes
1
answer
56
views
$S=\{x_n:x_n$ be a bounded string in $ \mathbb{R} \}$, with $k\in\mathbb{R}$, show that $S$ isn't compact with respect to sup norm.
Let $S=\{x_n:x_n$ be a bounded string in $\mathbb{R}\}$, with $k\in\mathbb{R}$ and $d((x_n),(y_n))=\sup\{|x_n-y_n|:n\in\mathbb{N}\}$. Show that $(S,d)$ is not compact.
I did it like this:
We know if $...
- 31
0
votes
1
answer
98
views
Continuous map from closed set to relatively compact set
Let $V$ be a complete normed space and let $f: X\to f(X)\subset X\subset V$ be continuous.
Can you find an example of a pair $f,X$ such that $X$ is closed in $V$ and $f(X)$ is relatively compact but ...
- 1,137
1
vote
0
answers
46
views
Prove that $A$ is relatively compact (or totally bounded) in $L^r(\Omega,\mu)$.
I encountered an exercise in my functional analysis course:
Let $(\Omega, \Sigma, \mu)$ be a measure space, and
$A\subseteq L^p (\Omega, \mu) \cap L^r (\Omega ,\mu)$, where $1\leq p <r <+\infty$...
- 716
0
votes
2
answers
65
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If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that $\sup_{x\in M} f(x)<a$
Let $(X,d)$ be a metric space, $M\subseteq X$ and $f: M \to \mathbb{R}$ continuous. Show that:
If $M$ is compact, then follows from $\forall x \in M | f(x) < a$ that
$\sup_{x\in M} f(x)<a$.
My ...
- 197
2
votes
1
answer
274
views
Open/compact sets in metric $d_1$ and $d_2$
In a set $X$ we consider two metrics $d_1$ and $d_2$.
We consider that the identity map $f:(X,d_1)\rightarrow (X,d_2)$ with $f(x)=x$ is continuous.
Which of the following statements are correct?
(a) ...
- 14.1k
5
votes
2
answers
301
views
Prove that $(K,\delta)$ is a compact metric space.
Let $(X, d)$ be a compact metric space. For $x ∈ X $ and $\epsilon > 0$, define
{$B_{\epsilon}(x) := {y ∈
X | d(x, y) < \epsilon}$}.
For $C ⊆ X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := ...
- 3,442
4
votes
1
answer
83
views
If $(X,d)$ is a compact metric space then is $A$ a closed subset of $K$?
Since $(X,d)$ is compact it is totally bounded so we will have a finite set $A_n$ such that $\displaystyle\bigcup_{x \in A_n} B_d\left(x,\frac{1}{n}\right)$ covers $(X,d)$. Now we take $A$ as the ...
- 3,442
4
votes
4
answers
211
views
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 ...
- 3,644
0
votes
1
answer
33
views
A metric space in which every point is contained in a ball whose closure is compact.
Suppose $X$ is a metric space such that for each $x\in X$, $\exists \epsilon_x>0$ such that $\overline{B(x,\epsilon_x)}$ is compact.Then show that this metric space is complete.
This is a problem ...
- 3,801
0
votes
0
answers
56
views
finite cover of clopen sets with diameter at most $\epsilon.$
Here is the problem:
Let $X$ be a compact metric space that is totally disconnected, and let $\epsilon > 0.$
(a) Show that $X$ has a finite cover $\mathcal{A}$ clopen sets with diameter at most $\...
user593471
0
votes
0
answers
45
views
finite cover of clopen sets.
Definition:
Suppose $\mathcal{A}$ and $\mathcal{B}$ are two covers of $X.$ We say that $\mathcal{B}$ refines $\mathcal{A}$ if each member of $\mathcal{B}$ is contained in some member of $\mathcal{A}....
user593471
0
votes
1
answer
230
views
Proving a property of a compact, totally disconnected metric space.
Here is the question:
If $X$ is a compact metric space that is totally disconnected, then for each $r > 0$ and each $x \in X,$ there is a clopen set $U$ such that $x \in U$ and $U \subseteq B_{r}(...
user593471
1
vote
2
answers
62
views
Converse statements regarding separable metric space
These statements:
a. Every infinite subset has a limit point
b. Separable
c. Has a countable base
d. Compact
are mentioned in the exercises of baby Rudin Chapter 2
I have proven with hints (in ...
- 141
4
votes
2
answers
589
views
Counterexamples to: If $f:X \to Y$ is continuous, $Y$ is compact, then $f^{-1}$ is continuous.
It is a theorem that if $f:X \to Y$ is a continuous bijection, $X$ is compact, then $g = f^{-1}$ is continuous. My professor asked us to find a counterexample to
If $f:X \to Y$ is continuous, $Y$ ...
- 24.4k
0
votes
2
answers
38
views
I am looking for an example of a sequence of points with the following conditions.
Usually if a set $\pmb K$ is compact, and every convergent subsequence of $(x_j)_{j=1}^\infty$, a sequence of points, converges to $x\in\pmb K$, then $(x_j)_{j=1}^\infty$ converges to $x$ as well.
...
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7
votes
1
answer
126
views
In a compact metric space, if we keep adding closed balls centered on boundary, do we always cover the entire space?
Let $X$ be a compact connected metric space, and let $W_1=B(x,r)$ denote the closed metric ball centered at $x\in X$ with radius $r$. We recursively define $W_{k+1}=W_k \cup B(y,r)$, where $y$ denote ...
- 73
0
votes
2
answers
41
views
Metric space compactness [duplicate]
Consider the space $C([0,1])$ equipped with the uniform norm.
Find a sequence of functions $\{g_n\}$ in $C([0,1])$ so that $\overline{\{g_n\}}$ is compact, but $g_n$ does not converge uniformly.
I'...
- 61
0
votes
2
answers
114
views
Is $\mathbb{Q} \cap (a,b)$ for irrationals $a,b$ is not compact in the $\mathbb{Q}$ with $d(x,y)=|x-y|$?
In my exam of analysis we ask to proof $\mathbb{Q} \cap (\sqrt{2},\sqrt{3})$ is not compact . i have few questions about this problem . i have a solution for the problem but beside my problem one of ...
7
votes
1
answer
2k
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Prob. 2 (d), Sec. 27, in Munkres' TOPOLOGY, 2nd ed: If $A$ is compact and $U$ is an open set containing $A$, then . . .
Here is Prob. 2, Sec. 27, in the book Topology by James R. Munkres, 2nd edition:
Let $X$ be a metric space with metric $d$; let $A \subset X$ be nonempty.
(a) Show that $d(x, A) = 0$ if and ...
- 27.2k
1
vote
1
answer
51
views
How i can show that this set is compact?
Let $(M,d)$ a metric space.
The following metric space $(M,d_A)$ is defined as:
$d_A(x,y)=\frac{d(x,y)}{1+d(x,y)}$
Show that if a set $F$ is compact on $(M,d_A)$ then is compact on $(M,d)$.
How i ...
- 1,342
0
votes
1
answer
137
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Family of Closed set of $X$ having finite intersection property with arbitary intersection is non empty Then $X$ is compact metric space
I wanted to prove Family of Closed set of $X$ having finite intersection property with arbitary intersection is non empty Then $X$ is compact metric space
I wanted to check my argument
On Contrary ...
- 6,773
0
votes
1
answer
139
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Isn't there any meaning to talk of open spaces, or of closed space?
From Rudin's Principles of Mathematical Analysis (p.37).
Theorem 2.33. Suppose $K\subset Y\subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$.
Immediately ...
- 801
0
votes
1
answer
57
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If Λ is a open subset of a metric space and $K⊆Λ_ε:=\{x:d(x,Λ^c)>ε\}$ is compact, is there a compact $L⊆Λ_ε$ s.t. $B_δ(x)⊆L$ for all $x∈K$?
Let
$(M,d)$ be a locally compact metric space
$\overline B_\delta(x)$ denote the closed $\delta$-ball of $x$ in $M$
$\Lambda\subseteq M$ be open
Now, let $\varepsilon>\delta>0$, $$\Lambda_\...
- 13.9k
0
votes
1
answer
41
views
Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.
This is Rudin's Real and Complex Analysis Problem 4.6.
Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.
Assume I have ...
- 2,754
2
votes
1
answer
282
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If A is a proper closed subalgebra of $C(X)$ ($X$ compact) and A $\subseteq I_{x_0}$ then $A = I_{x_0}$
Let $X$ be a compact set. Suppose $A$ is a proper, closed subalgebra of $C(X)$ that separates points.
Then $A \subseteq I_{x_0} = \{f \in C(X) : f(x_0) = 0\}$ for some unique $x_0 \in X$.
I ...
- 1,112
12
votes
2
answers
6k
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Showing that a totally bounded set is relatively compact (closure is compact)
I have been tasked with showing that for a metric space $(X,d)$, a subset $E \subseteq X$ is relatively compact $\iff$ $E$ is totally bounded. I believe I have shown the forward implication $(\...
- 1,669
1
vote
1
answer
34
views
For any compact $K \subset G$ open, how to show that there exist $R > 0$ s.t. $B(a;R) \subset G$ for all $a\in K$?
I think it's correct, at least in $\Bbb{R}^n$. It is a gap of another of my proof, and at first I didn't bother to show it and thought it is obvious. However when I tried to write an argument I was ...
- 3,208
6
votes
2
answers
4k
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Convergent Sequence + Limit is Compact using Sequential Compactness
Proposition: Let $(X,d)$ be a metric space and $\lbrace x_n \rbrace_{n=1}^\infty \subset X$ be a convergent sequence with $x_n \rightarrow x_0, n \rightarrow \infty$. Show that $K = \lbrace x_n \mid n ...
- 1,407
0
votes
2
answers
49
views
For each compact set $A$ in $\mathbb{R}$ its archetype $g^{-1}(A)$ is compact
Let $g: \mathbb{R} \rightarrow \mathbb{R} $ be a continuous and bijective function. prove: For each compact set $A$ in $\mathbb{R}$ its archetype $g^{-1}(A)$ is compact.
I know from my course that...
...
- 225
0
votes
1
answer
137
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Finding two compact sets.
Let $K$ be a compact set in a metric space $X$. Show that, if $K\subset U_1 \cup U_2$ for some open set $U_1$ and $U_2$, then there exist compact set $K_1$ and $K_2$ such that $K_1\subset U_1$, $K_2\...
- 1,866
1
vote
3
answers
877
views
Prove or counterexample: A closed and totally bounded set $A$ in a metric space must be compact.
I think for some time and now I agree that the statement is false. But I cannot prove or disprove the statement.
I know that compact $\Leftrightarrow$ complete+totally bounded, and also "bounded and ...
- 4,317
1
vote
1
answer
103
views
Compactness- Euclidean metric
Hi Could you help me to solve this question?
IF $||.||$ be any norm on $\mathbb{R}^m$ and let $B = \{ x \in \mathbb{R}^m : ||x||≤ 1 \}$ . Prove that $B$ is compact. Hint: It suffices to ...
- 33
0
votes
1
answer
77
views
About the uniform continuity of $\frac{\rho_F(x)}{\rho_F(x)+\rho_K(x)}$
Let $(X,d)$ be a metric space. For every non-empty subset $E$ of $X$, let $\rho_E: X \rightarrow\Bbb{R}$ be the function defined by $\rho_E(x) = \inf\{d(x,E)\,| \, y \in E\}$. This function is ...
- 864
3
votes
4
answers
2k
views
Showing subsets of $L^2=\{(x_n) : \sum_{n=1}^\infty x_n^2 < \infty \}$ are compact
Show whether the following subsets of $l^2$ are compact.
Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\},$$
equipped with the norm
$$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{...
- 910
2
votes
0
answers
105
views
Prob. 25(a), Chap. 4 in Baby Rudin: In $\mathbb{R}^k$, the sum of a compact set and a closed set is closed
Here is Prob. 25 (a), Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
If $A \subset \mathbb{R}^k$ and $B \subset \mathbb{R}^k$, define $A + B$ to be the set ...
- 27.2k
1
vote
0
answers
864
views
Prob. 10, Chap. 4 in Baby Rudin: Any continuous mapping of a compact metric space into any metric space is uniformly continuous
Here is Prob. 10, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Complete the details of the following alternative proof of Theorem 4.19: If $f$ is not ...
- 27.2k
1
vote
0
answers
170
views
Prob. 6, Chap. 4 in Baby Rudin: A function defined on a compact domain is continuous if and only if its graph is compact, but in what metric space?
Here is Prob. 6, Chap. 4 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
If $f$ is defined on $E$, the graph of $f$ is the set of points $\left( x, f(x) \right)$, for $x ...
- 27.2k
2
votes
1
answer
676
views
Example 4.21 in Baby Rudin: Why is the inverse of this function not continuous at this point?
Here's Theorem 4.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Suppose $f$ is a continuous 1-1 mapping of a compact metric space $X$ onto a metric space $Y$. ...
- 27.2k
8
votes
4
answers
2k
views
Does compactness depend on the metric? [closed]
If so, what are some examples of sets that are compact with respect to one metric but not compact with respect to another metric?
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