Tagged Questions
-2
votes
3answers
50 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 ...
2
votes
1answer
25 views
Every countably compact, scattered $T_3$-space is sequentially compact
A space $X$ is called scattered provided every non-empty subspace $Y$ has an isolated point( with respect to the subspace topology on $Y$).
How to prove that:
Every countably compact, scattered ...
4
votes
0answers
53 views
Is $X$ pseudocompact
The following example with a little modified from the handbook of set theoretic topology, Page 574:
Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
2
votes
2answers
52 views
Is my proof correct? (minimal distance between compact sets)
I'm working out the following problem form Ahlfors' Complex Analysis text:
"Let $X$ and $Y$ be compact sets in a complete metric space $(S,d)$. Prove that there exist $x \in X,y \in Y$ such that ...
5
votes
1answer
65 views
Understanding the topological space $\beta \omega$
It is from the Handbook of Set-theoretic topology:
We consider $\beta \omega$ as the set of all ultrafilters on $\omega$ with the topology which is generated by taking as a base all sets of the ...
3
votes
1answer
59 views
Compactness of the set of terms in a convergent (sub)net and its limit
In a Hausdorff topological space, let net $(x_d)_{d \in D}$ converge to $x$. The set
$$
(\cup_{d \in D} \{x_d\}) \cup \{x\}
$$
consisting of its terms and the limit need not be compact: in ...
15
votes
0answers
129 views
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic?
If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact the other one needs to be ...
5
votes
0answers
84 views
Is dependent choice necessary to prove every perfect compact Hausdorff space is uncountable?
The answer to Cardinality of a locally compact space without isolated point shows that AC is required to show that if $X$ is a compact Hausdorff space with no isolated points then $|X| \ge ...
3
votes
1answer
187 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 ...
-1
votes
1answer
140 views
Any net in $A\subseteq X$ has a cluster point in $X$. Is $\overline{A}$ compact?
Let $(X,\mathcal T)$ be a topological space, $A\subseteq X$ and any net in $A$ have a cluster point in $X$. Is $\overline{A}$ compact?
4
votes
2answers
95 views
How I can prove that: If $X$ is compact, then any map $f\colon X \to Y$ is proper?
A continuous map $f\colon X \to Y$ of locally compact spaces is called proper if for any compact $C\subset Y$ the preimage $f^{-1}(C)$ is compact. My question is: How I can prove that: If $X$ is ...
8
votes
1answer
100 views
One question related with sequential compactness
While trying to prove that a set is sequentially compact, I was suggested to prove by contradiction -- this is how it went, at least part of it:
Definition. We say that a set $A$ is sequentially ...
6
votes
3answers
138 views
Stone-Čech Compactification of the Natural numbers
I am trying to prove that if $U$ is contained in the Stone-Čech Compactification of the natural number ($\beta N$) that the closure of $U$ is open.
I have a really hard time with even understanding ...
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
49 views
Inverse limit of compact metric spaces
Does an inverse limit of compact metric spaces need to be metrizable? When it is an inverse limit of a countable inverse system I know it is metrizable (even without compactness). But what if the ...
3
votes
1answer
70 views
Tychonoff Theorem and the axiom of choice
How to show that
The Tychonoff Theorem and the axiom of choice are equivalent?
Here I want to collect ways to prove it.
Thanks for your help.
1
vote
1answer
61 views
If $X$ is complete and totally bounded, then $X$ is compact [closed]
Let $X$ be a metric space. Whar is your favorite way to show:
If $X$ is complete and totally bounded, then $X$ is compact?
Thanks for your help.
3
votes
2answers
50 views
A $T_2$ space is locally compact iff..
A $T_2$ space is locally compact iff it has a base $\beta $ s.t $\forall B\in \beta $ we have
$\bar B$ is compact.
My definition of locally compact is that $\forall x\in X$ has a compact neighborhood ...
6
votes
2answers
63 views
Question on compactification
I was studying for quals and had trouble with this question. Any help would be great, thanks.
A two-point compactifcation of a Hausdorff space $X$ is a compact Hausdorff space
$Y$ such that $X$ is a ...
4
votes
0answers
110 views
If $f: X \to Y$, when do we have $\beta Y \supset \overline{f(X)} = \beta X$?
Suppose that $X$ and $Y$ are Tychonoff spaces, denote by $\beta X$ and $\beta Y$ their Stone-Čech compactifications and let $f:X\to Y$ be a continuous map.
Using the embedding $Y\hookrightarrow\beta ...
8
votes
1answer
106 views
Clopen subsets of a compact metric space
I am aked to show that in a compact metric space we can find at most countably many subsets which are both: open and close. I would be grateful for your help.
3
votes
3answers
133 views
$f:X\to X$ is one-one and continuous on a compact space. Is $f$ surjective?
Let $(X,\mathcal T)$ be a compact Hausdorff topological space and $f:X\to X$ be one-to-one and continuous. Is $f$ surjective?
2
votes
2answers
32 views
Question about finite sets/compactness
I understand that every finite subset of a metric space is compact. But are there any topological spaces where finite sets are not compact? Is that even possible? I don't think it is but I just want ...
2
votes
1answer
43 views
On compact topological group
Must a compact topological group be metrizable? If not, is it separable?
Thanks for any help.
13
votes
10answers
998 views
How to prove $[a,b]$ is compact?
Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
3
votes
2answers
32 views
Is every countably compact space feebly compact?
A topological space is said to be feebly compact if every locally finite cover by nonempty open sets is finite.
Every compact space is feebly compact but how about countably compact spaces?
5
votes
1answer
157 views
Under what condition only does every compact subset of $X$ is closed implies $X$ Hausdorff?
It is trivial to see that:
If $X$ is Hausdorff, then every compact subset of $X$ is closed.
I am asking under what condition does the converse hold, i.e. when does
If every compact subset of $X$ is ...
1
vote
2answers
50 views
Hausdorff space and Cantor's intersection theorem
$X$ is a Hausdorff space, $C_i$ is a non-empty closed subset of $X$ and $C_{k+1}\subseteq C_k$ , show that $\displaystyle \bigcap_{i\in \mathbb{N}} C_i$ is compact.
I tried to prove by ...
4
votes
2answers
56 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?
3
votes
0answers
41 views
Another question in relation to Tychonoff theorem
Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here ...
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 ...
4
votes
2answers
90 views
Compact space and Hausdorff space
A continuous map from a compact space to a Hausdorff space is closed. Why this is true?
Help me please I want to learn why this is correct.
1
vote
1answer
49 views
Separating disjoint compact sets in Hausdorff space by using open sets
If $X$ is a Hausdorff Space and there are 2 disjoint compact sets $A,B\subset X$, we want to prove $\exists V,U \subset X, V \bigcap U=\emptyset $ S.T. $A\subset U, B\subset V$.($U,V$ are open sets)
...
2
votes
1answer
35 views
A restricted continuous map is a homeomorphism
Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists $ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
1
vote
1answer
54 views
Compactness and connectedness on $M_n(\mathbb R)$
Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices.
Which of the following are compact and which are connected?
a) The set of all invertible matrices
b) The set of all orthogonal ...
8
votes
6answers
284 views
Give an example of a simply ordered set without the least upper bound property.
In Theorem 27.1 in Topology by Munkres, he states "Let $X$ be a simply ordered set having the least upper bound property. In the order topology, each closed interval in $X$ is compact."
(The LUB ...
11
votes
1answer
111 views
Does there exist a topology for a set $X$ which is compact and Hausdorff?
For every set $X$ and every topology $\tau$ over $X$ we have that $\tau$ contains the trivial topology $\{ X, \emptyset\}$, which is compact, and is contained in the discrete topology $\{ S: S ...
1
vote
1answer
42 views
Pointwise Sup of continuous functions using Compactness
Consider two topological spaces $X$, $Y$. Assume $X$ is compact.
Question: Is it true that for any continuous function $f:X\times Y\rightarrow Z$ (where $X\times Y$ has the product topology and ...
4
votes
1answer
49 views
Compactification of a discrete space using ultrafilters.
I want to show for the collection of ultra filters on a (non-empty) set $A$, $U(A)$. That $U(A)$ is compact where the topology is derived from the base $U_B = \{F\in U(A)|B\in F\}$.
Seeing as $A$ can ...
0
votes
1answer
34 views
Metacompactness of the Euclidean space
Does anyone know how to prove that every Euclidean space is countably metacompact?
In particular, my interest is in $R^2$.
Thanks,
Shir
3
votes
1answer
54 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 ...
3
votes
1answer
109 views
Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?
My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
3
votes
1answer
69 views
Intuition behind compact subspaces of a metric space
I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following):
$E$ is compact in $X$ if for every open covering of ...
2
votes
1answer
72 views
Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.
I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification.
Does this hold in general?
I.e given a compact ...
4
votes
1answer
44 views
How to show this space $X$ is countably compact, first countable?
Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest
cardinal greater then $2^\omega$, equipped with the ordered topology
consisting of all ordinals of countable cofinality.
How to ...
4
votes
1answer
56 views
Is a minimal Hausdorff uniformity compact?
Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$,
$$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$
Is $(X,\mathcal D)$ compact?
0
votes
1answer
61 views
equivalence of compactness and countably compactness
Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
2
votes
1answer
35 views
exercises in compactness
I am working on some practice problems on Compactness. (Q.1.a Chapter 1.7 in Advanced Calculus, Folland)
The question is :
Give an example of :
a closed set $S\subset R\quad$ and a continuous ...
3
votes
1answer
28 views
No unbounded real continuous function on $X$ can be extended to a continuous real function on $\beta X$
By the Čech-Stone compactification theorem, I know that if $X$ is Tychonoff and $f:X\to [a,b]$ is continuous then $f$ can be extended to $\hat{f}:\beta X\to [a,b]$.
How can we show that no unbounded ...
1
vote
1answer
42 views
Show $\tau=\tau^*$ if $\tau^*\subset \tau$ [duplicate]
Let $(X,\tau)$ be compact and $(X,\tau^*)$ be a Hausdorff space. How can we show that $\tau=\tau^*$ if $\tau^*\subset \tau$?
