1
vote
2answers
55 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
1
vote
1answer
90 views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
3
votes
1answer
39 views

Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
3
votes
0answers
92 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathcal{Top}$, $k$-closed subset $Y\subset X$, means $u^{-1}(Y)$ is closed in $C$ for any $u: ...
1
vote
1answer
84 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
112 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
61 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
24 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
50 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
122 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
103 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
93 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
80 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
240 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
278 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
118 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
77 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
113 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
726 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
380 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} ...
9
votes
1answer
2k 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
242 views

Noncompact sequentially compact space

Have you an example of a noncompact sequentially compact space, without using ordinal?
3
votes
4answers
893 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
615 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
201 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? ...
9
votes
2answers
794 views

compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
12
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 ...