# Questions tagged [second-countable]

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

76 questions
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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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0answers
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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 ...
2answers
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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 ...
1answer
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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. " ...
1answer
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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 ...
2answers
141 views

### $\Bbb{R}^\omega$ 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 ...
2answers
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### Show that if a topological space has at most countable basis, then the space is separable and Lindelöf

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 ...
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1answer
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### Prove that $\mathbb{R}_l$ is not a second countable. ($\mathbb{R}_l$ are the real ones with the topology of the lower limit) [duplicate]

Prove that $\mathbb{R}_l$ is not a second countable. ($\mathbb{R}_l$ are the real ones with the topology of the lower limit) I have tried to reason for the absurd and suppose that $\mathbb{R}_l$ has ...
2answers
192 views

### Determine if $\mathbb{R}^{\omega}$ with the product topology is second countable

Determine if $\mathbb{R}^{\omega}$ with the product topology is second countable. I do not know if this space is second countable or not, I have noticed this post Product of infinite discrete space ...
1answer
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### Indiscrete topological space is second countable.

Is uncountable indiscrete topological space (X, tau )is second countale ? As far I know the only possible basis for the indiscrete topological space is X but since X is uncountable so its not second ...
1answer
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### Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].

Let $\mathbb R_l$ be $\mathbb R$ given the lower limit topology, i.e. the topology generated by $\{[a, b) \subseteq \mathbb R|a<b\}$. Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has ...
0answers
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### Prove that $\mathbb R_l$ $\times$ $\mathbb R_l$ given the box topology has a countable dense subset.

Let $\mathbb R_l$ be $\mathbb R$ given the lower limit topology, i.e. the topology generated by $\{[a, b)\subseteq \mathbb R|a<b\}$. Prove that $\mathbb R_l$$\times$$\mathbb R_l$ given the box ...
0answers
252 views

### Separable, metrizable, and second countable spaces

A space $X$ is second countable if it has a countable basis. We say that a space $X$ is separable if there is a set $Y\subseteq{X}$ such that $Y$ is countable and dense in $X$. Show that if $X$ is a ...
1answer
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### $X$ is a compact Hausdorff space prove that the diagonal $\Delta$ in $X \times X$ is a $G_{\delta}$ -set

The problem says: Suppose that $X$ is a compact Hausdorff space. If there exists a continuous function $f \colon X \times X \to \mathbb{R}$ such that $f (x, y) = 0 \iff x = y$, prove that the ...
1answer
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### Countable and dense Baire subsets

I'm having trouble with an excercise, which says: Let $A\subset \mathbb{R}$ be a Baire space. Then $A$ can't be countable and dense simultaneously. I think I have found a proof, but it doesn't ...
1answer
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### basis of a second countable, topology [closed]

Prove that every basis of a second countable space contains a countable subfamily which is also a basis. *I try with the intersection of the bases, the open ones of the finite with those of the other,...
3answers
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### Second Countability Proof

Let $F: X \rightarrow Y$ be a continuous open map. Show that if X satisfies the first or second countability axiom then $F(X)$ satisfies the same condition. Attempt at proof: Suppose X is second ...
3answers
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### If $\sqcup_{i\in A} X_i$ is second countable and $X_i$ is second countable, then $A$ is countable. Is this true?

This is basically a general statement adapted from Lee's Topological Manifold Ex.3.44 in the text. $\{X_i\vert i\in A\}$ is a set of $n-$manifolds. $\sqcup_{i\in A} X_i$ is $n-$manifold endowed with ...
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1answer
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### Does having a countable subbasis also satisfy the second axiom of countability? [duplicate]

My guess is "yes". Unfortunately, the course I've taken this semester on "point-set topology" did not cover the concept of subbasis, however I have a few words on it in mind, subbasis unlike basis ...
2answers
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### $X$ top space with countable basis $B = \{B_n\}_{n\in \mathbb{N}}$. Then exists countable subset $D\subseteq X$, $\overline{D} = X$

Let $X$ be a topological space with a countable basis $B = \{B_n\}_{n\in \mathbb{N}}$. Show that exists a countable subset $D\subseteq X$ such that $\overline{D} = X$. Well, I've thought a lot about ...
1answer
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### Some question about first (or second) countable - first (second) “countabilization”

Given $X$ be a topological space (or maybe a topological group), I am curious about is there a coarser or finer topology of $X$ or some quotient space such that it become first (or second) countable. ...
1answer
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### Every Riemann Surface has a countable basis for its topology

In the book Introduction to Teichmüller Spaces, by Taniguchi and Imayoshi, we have the following definition for a Riemann Surface: At the following pages, the authors make a remark recalling some of ...
3answers
262 views