Tagged Questions
2
votes
1answer
41 views
Proof that the interior of any union of closed sets with empty interior in a compact Hausdorff space is empty
The question is pretty much in the title, I need to show that given $X$ is a compact Hausdorff space and $\left\{ A_n\right\}_{n=1}^\infty$ is a collection of closed subsets of $X$ each with empty ...
4
votes
1answer
27 views
Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $
I'm trying to find a sufficient and necessary condition for a subset to be compact in $\mathbb{R} $ when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, ...
-2
votes
3answers
52 views
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set.
(Can we use ...
3
votes
1answer
192 views
One point compactification of $[0,1] \times [0,1)$
Let $X = [0,1] \times [0,1) \subset \mathbb{R}^2$. I've already proven that this space is locally compact and found its one-point compactification but now I am stuck on the following; Let $Y = X \cup ...
0
votes
1answer
55 views
A question on the compact subset of $[0,1]$
Let $S=\{K \subseteq [0,1]: K \text{ is compact and uncountable } \}$. How to see that $|S|=\mathfrak c$?
Thanks for your help.
1
vote
2answers
45 views
Finding a bounded, non-compact set of functions $f:[0,1]\to\Bbb R $
Consider the metric space $(X, d)$ given by $$X = \{\text{all continuous functions}\,f:[0,1]\to\Bbb R\}$$ with $$d(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|.$$ Find with proof a set $A \subseteq X$ with ...
6
votes
0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
4
votes
2answers
58 views
one point compactification
I am asked to describe the one point compactification of $(0,1) \cup [2,3)$ of $\Bbb R$ and if I'm not mistaken it is just a circle union the closed set [2,3] correct? Am I missing something?
2
votes
1answer
91 views
How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?
A topological space $X$ is called sequentially compact if every sequence of points in $X$ has a subsequence that converges to a point in $X$. I know it's very similar to Bolzano–Weierstrass theorem ...
2
votes
2answers
63 views
Compact inclusion in $L^p$
Is it true that there is a compact inclusion from $L^p$ to $L^q$ whith $q<p$?
What is the counterexample if what I said is wrong?
Thank you.
6
votes
1answer
65 views
Tight Probability on Hilbert space
I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$
And for any ...
1
vote
2answers
51 views
Let $ f:(X, d)\mapsto (X, d ) $ be a mapping on compact metric space with $ d (f (x), f (y))<d (x,y) $for $ x\ne y $
I prove that $ f $ has a fixed point. My question is whether the point is unique and the mapping $ f $ is continuous.
3
votes
1answer
55 views
How to show that a continuous map on a compact metric space must fix some non-empty set.
Suppose $(X,d)$ is a compact metric space and $f:X\to X$ a continuous map. Show that $f (A)=A$ for some nonempty $A\subseteq X.$
I start this by supposing that $A_0:=X$ and $A_{n+1}:=f(A_n)$ for ...
1
vote
1answer
24 views
Prove that $ A\subset \ell_1 $ is compact iff $A$ satisfies the following property
$A$ is compact iff $ A $ is bounded and, given $\epsilon > 0$, there exists $ n_0$ such that $ \sum_ {k=n}^\infty |x_k|\le\epsilon $ for all $n \geq n_0 $ and for all $ x\in A $.
To prove ...
1
vote
1answer
104 views
SHOW that there are infinitely many equivalence classes of formulas
Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the
language of $\mathcal{Q}$ and let $T$ be the ...
0
votes
2answers
25 views
Prove that $D ⊂\Bbb R^{n}$ is compact iff whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$
Prove that $D ⊂\Bbb R^{n}$ is compact if and only if whenever {$C_{α}$} is a collection
of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$ , there is a finite
subcollection satisfying ...
3
votes
1answer
148 views
closed bounded subset in metric space not compact
Let $\ell^{\infty}$ be the space of bounded sequences of real numbers, endowed with the norm $\|\mathbf x\|_\infty=\sup_{n\in N}|x_n|$, where $\mathbf x=(x_n)_{n\in\Bbb N}$.
Prove that the closed ...
9
votes
1answer
84 views
Ideal in compact Hausdorff space
This is exercise 70, chapter 4. from Folland (page 142)
Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of
$C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
3
votes
1answer
77 views
$W^{1,p}$ compact in $L^\infty$?
Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
3
votes
1answer
83 views
A locally compact subset of a locally compact Hausdorff space is locally closed
Let $X$ be a locally compact and Hausdorff space. Show that if $Y \subset X$ is locally compact, then $Y$ is locally closed, in essence $Y$ is an open subset of $\overline{Y}$, where $\overline{Y}$ ...
4
votes
2answers
199 views
Closed Bounded but not compact Subset of a Normed Vector Space
Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is
$||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$.
Prove that $B'(0,1) = \{x \in l^\infty ...
1
vote
1answer
86 views
Is this space countably compact
Let $X$ be a Tychonoff countably compact space and $A$ is a subapce of $X$ such that for any countable $B \subset A$ we have $\overline{B} \subset A$. My question is this:
Is this subspace $A$ ...
0
votes
2answers
107 views
To find given set is open, closed, bound, closure, interior
Consider the set $S = \{x \in \mathbb R^2 :-1 < x_i \leq 1,\; i = 1,2\}$. Is $S$ open? Is it
closed? Is it bounded? Is it compact? Find the closure, interior, and boundary of S.
Aside from the ...
9
votes
3answers
244 views
A question on a compact space
Show:
If the closure of every discrete subset of a space is compact then the whole space is compact.
Thanks advance:)
2
votes
3answers
89 views
how to show that the compactness of set implies closedness?
Let $S$ be a subset of $\mathbb R^n $. Then the following statements are equivalent
a) $S$ is compact
b) $S$ is closed and bounded
c) Every infinite subset $S$ has an accumulation point in ...
1
vote
2answers
90 views
Proof: if $K\subset M$ is compact and $A\subset K$ is closed, then $A$ is compact [duplicate]
Possible Duplicate:
Compactness is closed-hereditary - significance of closed property?
Proposition let $(M,d)$ be a metric space where $K\subset M$ is compact and $A\subset K$ is closed. ...
4
votes
1answer
104 views
Prove that the following subset is compact
Let $A \subset \ell^p$, where $1 \le p \lt \infty$. Suppose the following conditions are true:
1) $A$ is closed and bounded
2) $\forall \epsilon \gt 0, \: \exists \: N \in \mathbb{N}$ ...
0
votes
1answer
295 views
How to prove that the closed convex hull of a compact subset of a Banach space is compact?
Can anyone help me with this problem?
Prove that if $K$ is a compact subset of a Banach space $X$, then the closed convex hull of $K$ (that is, the closure of the set of all elements of the form ...
0
votes
2answers
95 views
Compactness of union and intersection?
Consider a compact subset $A\subseteq \mathbb{R}$ together with a closed subset $B\subseteq \mathbb{R}$. Show that $A\cap B$ is compact. Is $A\cup B$ also compact?
3
votes
2answers
647 views
Prove: Every compact metric space is separable
How to prove that Every compact metric space is separable$?$
Thanks in advance!!
8
votes
2answers
553 views
If $A$ and $B$ are compact, then so is $A+B$.
This is an exercise in Chapter 1 from Rudin's Functional Analysis.
Prove the following:
Let $X$ be a topological vector space. If $A$ and $B$ are compact subsets of $X$, so is $A+B$.
My guess: ...
3
votes
1answer
121 views
Describe all the compact subsets of this space
Consider the topological space $(X,\mathscr{U})$, where $X=\mathbb{R}^2$ and the topology $\mathscr{U}$ is generated by the collection of sets $\{(0,0)\}\cup \{I_a\}$ where $I_a$ are the open ...
3
votes
3answers
122 views
Existence of such points in compact and connected topological space $X$
Let $X$ be a topological space which is compact and connected.
$f$ is a continuous function such that;
$f : X \to \mathbb{C}-\{0\}$.
Explain why there exists two points $x_0$ and $x_1$ in $X$ such ...
1
vote
3answers
209 views
compact metric space problem
Let $(X,\tau)$ a compact metric space and $\{ U_i : i \in I \}$ an open cover of X.
Show that there is $r>0$ such that for all $a \in X$ there is an $i \in I$ such that $B_{r}(x) \subseteq U_{i}$.
...
1
vote
1answer
449 views
An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.
Why is this true?
I think I can find a countable union of compact sets $\cup_{k=1}^\infty X_k$ such that $\cup X_k \subseteq U$ and the lebesgue measure of $U \setminus \cup X_k$ is zero.
(for any ...
3
votes
1answer
169 views
Set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ is not separable
I want to show the set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ (where $\beta \mathbb{N}$ is all ultrafilters on $ \mathbb{N}$) is not separable. I know we can take as a base of $\beta ...
