# Questions tagged [second-countable]

For questions about second-countable topological spaces, i.e., space with countable base.

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### Second Countability Implies Separability

Munkres states explicitly that: Theorem 30.3: Suppose that $X$ has a countable basis. Then: (b) There exists a countable subset of $X$ that is dense in $X$. The following proof he gives is ...
1answer
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1answer
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### Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal?

Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal? b) If X is regular, and if each point of X has a fundamental countable neighborhood system and is ...
2answers
29 views

### Is this topology separable, first countable, second countable?

Consider a set $E$ with some point $p \in E$ and the topology: $T = \left \{ \emptyset \right \}\cup \left \{ C \subset E: p \in E \right \}$. Is this topology, separable, first countable or second ...
0answers
45 views

### Examples of topologies in which all open sets are countable union of base elements, but the topology itself doesn't have countable bases.

In second countable topology(i.e. has countable bases), obviously all open sets are countable union of base elements. But the inverse seems not true. The sorgenfrey line $R_l$ seems to be an example. ...
1answer
51 views

### Second countable space implies numerable subbasis [closed]

If a topological space (X,t) has a numerable basis, implies X has a numerable subbasis?
1answer
80 views

### Manifold has a countable cover by compact sets.

I was confused by Boothby's proof that a Manifold had a countable cover of compact sets. In his proof he says that as $M$ has a countable basis of open sets, we may assume that $M$ has a countable ...
1answer
44 views

### Second countability of compact open topology

Let $X, Y$ be topological spaces and $C(X,Y)$ the set of continuous functions with the compact open topology defined as having the subbasis consisting of all subsets $C(K, U)$ of functions $f$ for ...
2answers
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1answer
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### Second countable and sequentially compact topological spaces are compact. Proof [duplicate]

I can’t prove that a space having a countable basis and being sequentially compact is indeed compact. Can anyone help me with the prove? Thanks!
1answer
82 views

### How can I prove that compact subset iff sequentially compact in second countable topological spaces?

Let $X$ be a second countable topological space. Then $A \subset X$ is a compact subset of $X$ if and only if every sequence $\{x_{n}\}_{n \in \mathbb{N}} \subset A$ has a convergent sub-sequence with ...
2answers
46 views

### Is every compact subset of a second countable space closed? [closed]

I know that this is true for Hausdorff spaces and metric spaces, which are Hausdorff spaces, but I can’t prove it for second countable spaces. Is it even true? Thanks!
1answer
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0answers
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### Countable base for the one point compactification topology on $\mathbb{C}$

I see similar results on this site, but I am unable to find the explicit base, $$\{B(x,1/n):x\in \mathbb{C}, x=a+ib,a,b\in \mathbb{Q},n\in \mathbb{N}\}\cup \{0\}^c$$. Will this work?
1answer
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### $\mathbb{R^n}$ Second Countable

Lemma: Let U be an open subset of $\mathbb{R}^n$ and $x \in U$ then there exists a ball with rational center $(\mathbb{Q}^n$ and rational radius containing x, that is contained in U. What I want to ...
3answers
107 views

### Proving that $\Bbb Q$ is dense in $\Bbb R$

I would like a hint for the following: $\forall \epsilon >0$ $\forall x\in \mathbb{R}$ $\exists y \in \mathbb{Q}$ $:$ $|x-y|< \epsilon$. The idea I have is to break it into the following cases:...