0
votes
1answer
55 views

Does sequential compactness imply countable compactness?

Let $X$ be a topological space which is sequentially compact. Does this imply that $X$ is countably compact? Thank you!
2
votes
1answer
70 views

Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
0
votes
1answer
54 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
1
vote
1answer
18 views

How can a bounded subspace of the left order topology be compact?

I want to show that every bounded set equipped with the left order topology is compact. This is a statement I found on a wikipedia page and appearently it is lifted from the book Counterexamples 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
0answers
108 views

Weak Hausdorff space not KC

I am stuck with a problem in general topology. First of all, recall that a space $X$ is KC if every compact subset of $X$ is closed, and is weak Hausdorff if for all $u:K\rightarrow X$ continuous ...
3
votes
1answer
89 views

The product of limit point compact Hausdorff spaces is not limit point compact

Let $X, Y$ be limit point compact Hausdorff spaces (to be clear, a space is said to be limit point compact if every infinite subset of it has a limit point). Is it true that $X \times Y$ is limit ...
3
votes
1answer
82 views

Need a unique convergence (UC) space's Alexandrov extension be a UC space?

Background Say a topological space $X$ is a unique convergence (UC) space iff every sequence of points of $X$ converges to at most one point of $X$; a unique convergent clustering (UCC) space iff ...
0
votes
1answer
75 views

Compact spaces where not all compact subsets are closed

A topological space $(X,\tau)$ is called $C-C$ iff the closed sets in $X$ coincide with the compact sets in $X$. A topological space is called a $US$-space provided that each convergent sequence has ...
0
votes
2answers
190 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 ...
-1
votes
1answer
226 views

A simple example of Lindelöf space.

Somebody can to give me a simple example of Lindelöf space? Note. Lindelöf space is a topological space in which every open cover has a countable subcover.
8
votes
3answers
109 views

Is this kind of space metrizable?

It has a nice result from Tkachuk V V. Spaces that are projective with respect to classes of mappings[J]. Trans. Moscow Math. Soc, 1988, 50: 139-156. If the closure of every discrete subset of a ...
4
votes
0answers
75 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 $\{H_\alpha: \alpha < \kappa\}$ ...
0
votes
1answer
106 views

Unique nearest point in epsilon neighborhood of compact real manifold?

I have to proof the following assertion: Let $X$ be a compact submanifold of $\mathbb{R}^n$ and $\mathcal{U}^\varepsilon=\{p\in\mathbb{R}^n\;:\; |p-q|<\varepsilon \text{ for some }q\in X\}$. Then ...
8
votes
7answers
592 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 ...
2
votes
1answer
290 views

Compactness, Local Compactness, Completeness

Clearly, every compact metric space is locally compact. I always get confused when completeness is introduced into the picture. Which of the following are true? What are some easy counterexamples to ...
4
votes
1answer
48 views

Questions about an example

Recently, I met an example. I have two questions about the example: Why the author said, because $z \notin A$, then $z$ is not in the closure in $\beta \mathbb{R}$ of $A \cap (\beta \mathbb{R} ...
7
votes
1answer
1k views

Intersection of finite number of compact sets is compact?

Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of ...
5
votes
2answers
224 views

Noncompact sequentially compact space

Have you an example of a noncompact sequentially compact space, without using ordinal?
2
votes
4answers
820 views

Every compact subset must be closed?

This is an exercise from a topological book. In $T_1$ space, every compact subset must be closed? For any two compact subset, their intersection must be compact? Thanks for any help:)
2
votes
1answer
535 views

Intersection of countable set of compact sets

I am asking whether a specific construction is a counterexample to Theorem 2.36 in Rudin's "Principles..." book (3rd Ed.), which reads, 2.36 Theorem If $\{K_{\alpha}\}$ is a collection of compact ...
3
votes
3answers
189 views

Discontinuous function sending compacts to compacts

I know that the condition that $f(X)$ is compact if $X$ is compact should not be sufficient to say that $f$ is continuous, but I can't come up with an example of such discontinuous $f$. What is it? ...
11
votes
5answers
2k views

A compact Hausdorff space that is not metrizable

Is there an example of a compact Hausdorff space that is not metrizable? I was thinking maybe the space of continuous functions $f: X \rightarrow Y$ between topological spaces $X, Y$, might work, but ...