4
votes
1answer
42 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
1
vote
1answer
47 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
0
votes
1answer
48 views

Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
1
vote
1answer
54 views

Using the compactness theorem

I am working through problems which ask you to apply the compactness theorem (from propositional logic) to problems. How would you go about solving this one? Let $\mathbf{L}$ be an arbitrary ...
0
votes
0answers
29 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
0
votes
1answer
33 views

Compact and connected

Let $\mathbb{J} :=\{1/n: 0< n\in \mathbb{Z}\}$ Let $T_{ir}$ be topology of $\mathbb{R}$ generated by $$\{(a,b)\subset \mathbb{R}:a<b\}\cup\{(a,b) \setminus \mathbb{J}\subset ...
0
votes
2answers
31 views

Is sequence convergent in subspace of compact metric space?

Problem is as follow. Let X be a compact metric space and A be a closed subset of X. Prove that every sequence in A has a convergent (note: convergent in A) subsequence. It is from my note. My ...
1
vote
3answers
96 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
1
vote
1answer
59 views

Closed subsets of compact sets are compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (1) Prove this using the definition of compactness. Can somebody prove it? I think we should select a open cover of S ...
0
votes
1answer
40 views

Topological contraction on compact spaces

This is a follow up question. You can see the original here. I have the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a ...
4
votes
2answers
98 views

Contraction of compact sets

I am trying to solve the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a non-empty set $A\subset X$ such that $f(A)=A$. ...
2
votes
3answers
70 views

On compact sets

Let $A$ be a subset of $\mathbb R$ with more than one element. Let $a\in A$. If $A\setminus \{a\}$ is compact, then $A$ is compact. every subset of $A$ must be compact. $A$ must be a finite set. $A$ ...
2
votes
0answers
33 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
2
votes
2answers
73 views

Proving the set $C = \{\,x \in \mathbb R^n : \sum x_i = 1, x_i \in [0,1]\,\}$ is compact.

Proving the set $C = \{\,x \in \mathbb R^n : \sum_{1}^n x_i = 1, x_i \in [0,1]\,\} \subseteq \mathbb R^n$ is compact. Alright: I can use the Heine-Borel theorem to prove this, therefore all I need to ...
1
vote
1answer
32 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
4
votes
1answer
48 views

Topological space in which there are no close and compacts subsets (except for the empty set)

Any example of those topological spaces? I cant think of no one :S I think it must be infinite and it must not be T2, but no idea how to find one.
4
votes
1answer
132 views

Stone-Čech compactification of discrete space

Let $X$ be a discrete space and $\beta X$ its Stone-Čech-compactification, given by $\overline{\iota(X)}$ where $$ \iota:\ X \to \prod_{f \in C(X,[0,1])} [0,1],\quad x \mapsto (f(x))_{f \in ...
0
votes
1answer
105 views

Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...
0
votes
1answer
210 views

$X$ normed linear space separable $\Longleftrightarrow$ $\exists K \subset X$ compact s.t. $\overline{ \text{span}\{K\}}= X$

Let $X$ be a normed linear space. Show that $X$ is separable if and only if there is a compact subset $K$ of $X$ for which $\overline{ \text{span}\{K\}}= X$ I can't figure out how to solve this ...
0
votes
1answer
54 views

Sorgenfrey's topology- compact subsets

Consider the topological space $(R,\tau_{sor})$ Are the atoms $\{x\}$ where $x\in X $ compact in Sorgenfrey's topology? And in the usual topology? To say if $\{0\}\times [0,\infty)$ is compact in ...
2
votes
1answer
62 views

One point compactification is second contable

$(X,\tau)$ is a local compact, second countable Hausdorff space s.t. $ X = \cup_n K_n$ for countably many compact sets in $X$. Then $\infty$ has a countable neighborhood basis of open sets where ...
1
vote
3answers
199 views

For a compact covering space, the fibres of the covering map are finite.

I am stuck on the following exercise: Let $Y$ be a compact topological space, and $p:\ Y\ \longrightarrow\ X$ a covering map. Show that for every $x\in X$ the fibre $p^{-1}(x)$ is finite. Any ...
4
votes
1answer
74 views

Regarding compactness of a space

I am trying to solve the following problem: Let $X$ be a metrizable topological space. Prove that the following statements are equivalent. (a) $X$ is compact (b) $X$ is bounded with respect to ...
1
vote
0answers
84 views

Is this proof correct about compact sets inside open sets?

I've been solving the following problem: "If $U\subset\mathbb{R}^n$ is open and $C\subset U$ is compact, show that there is a compact set $D$ such that $C\subset \operatorname{int}(D)$ and $D\subset ...
1
vote
1answer
118 views

Compact Subsets of $C[a,b]$

Consider the set $G = \lbrace f \in C\left[a,b\right] : |f(x)| \le |g(x)|,\ \forall x \in [a,b] \rbrace$ Find all values of $g$'s for which $G$ is a compact subset of $C[a,b]$ with the max norm. ...
3
votes
2answers
68 views

Compactness and Hausdorffness

Is it possible to have a topological space having all compact subsets closed, but the space itselt is not Hausdorff? I couldn't find any counterexample. I tried $\mathbb R$ with cofinite, point ...
0
votes
4answers
183 views

Uncountable open cover of $\mathbb{R}$

The question posed to me is to find an uncountable open subcover of $\mathbb{R}$ such that it has no finite subcover, but I can't even think of a way to define an uncountable open cover.
0
votes
0answers
131 views

If $B$ is sequentially compact and $A \subseteq B$ is closed, then $A$ is sequentially compact

$DEF:$ A set $X$ is sequentially compact if $\{x_k\}_{k \geq 1} $ is a sequence and $x_k \in A$ for each $k$, then there is a point $x \in A$ so that $x$ is a limit point of $\{x_k\}_{k \geq 1} $ ...
0
votes
1answer
64 views

locally compact and compatification

As we know a topological space $X$ is said to be locally compact at a point $ x \in X $ if $x$ has a compact neighbourhood in $X$. $X$ is called locally compact if it is locally compact at every ...
3
votes
1answer
64 views

Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ has a finite subcovering.

Let $H \subseteq \Bbb R^n$. Prove that $H$ is compact $\iff$ every cover $\{E_{\alpha}\}_{\alpha \in A}$ where $E_{\alpha}$'s are relatively open in $H$ has a finite subcovering. $\bf{Solution \ ...
3
votes
2answers
67 views

Prove that $H$ is a finite set.

Let $H$ be compact in $\Bbb R^n$ Also assume that for every $x\in H$ there is an $r=r(x)$ such that $B_r(x)\cap H=\{x\} $ Prove that $H$ is a finite set. Solution: Since $H$ is compact, ...
0
votes
0answers
56 views

Is the following space $\sigma-$compact?

Let $X$ and $Y$ be two $T_{2}$ spaces, such that $Y$ is $\sigma-$compact. Let $f:X\to Y$ be an open continuous surjective map. Is $X$, $\sigma-$compact?
0
votes
2answers
227 views

locally compact Hausdorff space which is not second-countable

I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't ...
2
votes
1answer
75 views

Check my answer: H is compact $\iff$ every cover {${E_\beta}$}$_{\beta \in A}$ of $H$ has a finite subcovering.

Question: Let $H ⊆ \Bbb R^n$ Prove that H is compact $\iff$ every cover $\{{E_\beta}\}_{\beta \in A}$ of $H$, where $E_\beta$ 's are relatively open in $H$, has a finite subcovering. Solution: ...
1
vote
1answer
93 views

$\sum c_k^2<\infty$ then $A=\{\sum_{k=1}^{\infty} a_ke_k :|a_k|\leq c_k \}$ is compact

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal set in a Hilbert space $H$. If $\{c_k\}_{k=1}^\infty$ is a sequence of positive real numbers such that $\sum c_k^2<\infty$, then the set: ...
1
vote
1answer
98 views

Proper map, what's wrong?

"A map $f$ from $\mathbb R^2$ to $\mathbb R^2$ is proper if the full preimage of every compact set under $f$ is compact. Prove that every complex polynomial $f$ regarded as a self-map of the plane of ...
1
vote
1answer
279 views

Why is the inverse image of a compact set under a special sort of function compact?

Let $f$ be a continuous closed function from $X$ to $Y$ where $X$ and $Y$ are topological spaces. (Closed means that for any closed set $C$, $f(C)$ is also closed). Suppose that for any $y$ in $Y$, ...
5
votes
1answer
101 views

If $A$ is a compact set then so is $A'$?

Let $X$ be a metric space with a metric $d$ and let $A$ be a compact subset of $X$. Show that $A'$ is compact where $A'$ is a derived set of $A$. I am done $A'$ is closed and bounded. But we know ...
2
votes
1answer
280 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
98 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)\,|\, ...
-1
votes
3answers
120 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 ...
6
votes
1answer
520 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
60 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
312 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 ...
7
votes
0answers
106 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 ...
6
votes
2answers
300 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
225 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
166 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.
7
votes
1answer
99 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
122 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.