# Questions tagged [second-countable]

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

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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!
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### 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 ...
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### 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!
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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?
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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 ...
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### 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:...
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### Proving that $\Bbb Q^n$ is dense in $\Bbb R^n$

Lemma: $\forall \varepsilon >0$ $\forall x \in \mathbb{R}^n$ $\exists$ $y \in \mathbb{Q}^n$ so that $\|x-y\| < \varepsilon$. I'm pretty sure that I should be using the fact that the rationals ...
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### $L^{p}$ is separable for second countable space

I showed that $L^{p}$ is seprable when $X$ is a topological space with countable base and $\mu$ a radon measure on $T$. Do you think it's the right assumptions ? I mean does it exist a counter ...
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### Are countable topological spaces second-countable?

Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other ...
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### Second countable CW complexes

When exactly is a CW complex second-countable?
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### Adjunction spaces of second countable, locally compact, Hausdorff spaces

Suppose that $X$ and $Y$ are second countable, locally compact, Hausdorff spaces. Let $Z$ be a closed subspace of $X$ and suppose that $f : Z \to Y$ is an injective and open continuous map (hence a ...
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### The continuous image of a First Countable Space need not be First Countable (Willard 16.B.1)

In Stephen Willard's General Topology appears the following exercise: A quotient of a second countable space need not be second countable (for each $n\in \mathbb{N}$, let $I_n$ be a copy of $[0,1]$ ...
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### Check properties of the topology $\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\}$

Consider the topological space $(\mathbb{R}, \mathcal{T}:=\{U \subseteq \mathbb{R} \mid \forall x \in U: \exists \epsilon > 0: [x, x + \epsilon[ \subseteq U\})$. Is this space separable? Is it ...
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### Does partition of unity implies second countable?

Reading the definition of partition of unity: Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\Phi$ of $C^\infty$ functions $\varphi$ defined in an ...
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### A space is metrisable if and only if it admits a countable basis

I have a topology question about metrisable and countable basis. Please indicate whether the argument below is true or not: "A space is metrizable if and only if it admits a countable basis. &...
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### Compact T2 space with separable C(X) is second-countable

I am trying to prove this theorem: Let X be a compact Hausdorff space, such that $\mathbf{C}\left(X\right)$ is separable, then X is second-countable. I found a sketch of the proof here, but I am ...
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### $\Bbb{R}^\omega$ with the Uniform Metric is not 2nd Countable

I'm trying to show that $\Bbb{R}^\omega$ with the uniform metric is not 2nd countable or separable (either one will work, since they are equivalent for metric spaces). My idea was to transfer the ...
Let $(X,\mathcal{T})$ be a topological space. Show that if $(X, \mathcal{T})$ has countable base, it is separable (a) and Lindelöf (b) My attempt: Let $\mathcal{B}$ be a countable basis of the ...
I think that the following claim is wrong, but I could not come up with a counter example: Let $X$ be a compact, Hausdorff, second countable space. Assume that we have the following process: Let \$...