Questions tagged [second-countable]

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

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Okay… but what does “second-countable” actually mean?

A topology is a pair of sets $(X,\tau)$ such that $\tau \subseteq \mathcal{P}(X)$. $X,\emptyset \in \tau$. $\forall S\subseteq\tau.\bigcup S \in \tau$ $\forall A,B\in \tau.A \cap B \in \tau$ If $\...
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40 views

Prob. 16 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The product space $\mathbb{R}^I$, where $I = [0, 1]$, is separable

Here is Prob. 16 (a), Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that the product space $\mathbb{R}^I$, where $I = [0, 1]$, has a countable dense subset. My Attempt: ...
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26 views

Prob. 15, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The subspace $\mathscr{C}(I,\mathbb{R})$ of $\mathbb{R}^I$ with uniform metric is separable

Here is Prob. 15, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Give $\mathbb{R}^I$ the uniform metric, where $I = [0, 1]$. Let $\mathscr{C}(I, \mathbb{R})$ be the subspace ...
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1answer
28 views

Prob. 12, Sec. 30, in Munkres' TOPOLOGY, 2nd edition: Every continuous open image of a first / second countable space is also first / second countable

Here is Prob. 12, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Let $f \colon X \rightarrow Y$ be a continuous open map. Show that if $X$ satisfies the first or the second ...
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1answer
21 views

Is the product space of a non-second-countable space and a second-countable space non-second-countable?

I am asked if $\mathbb{R} \times [0, 1]$ would be a manifold if $\mathbb{R}$ is defined with lower limit topology and $[0, 1]$ is defined with subspace topology for standard topology. Although $[0, 1]$...
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1answer
36 views

Prob. 6, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable

Here is Prob. 6, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R}_\ell$ and $I_0^2$ are not metrizable. My Attempt: First, we consider $\mathbb{R}_\ell$. ...
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23 views

Prob. 5 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizable Lindelof space has a countable basis

Here is Prob. 5, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: (a) Show that every metrizable space with a countable dense subset has a countable basis. (b) Show that every ...
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36 views

Prob. 5 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every separable metric space is second-countable

Here is Prob. 5, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: (a) Show that every metrizable space with a countable dense subset has a countable basis. (b) Show that every ...
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1answer
45 views

Prob. 4, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every compact metrizable space has a countable basis.

Here is Prob. 4, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that every compact metrizable space $X$ has a countable basis. [Hint: Let $\mathscr{A}_n$ be a finite covering ...
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1answer
37 views

Prob. 2, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every basis of a second-countable space contains a countable basis

Here is Prob. 2, Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ has a countable basis $\left\{ B_n \right\}$, then every basis $\mathscr{C}$ for $X$ contains a ...
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100 views

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
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1answer
49 views

Radially open set topology is separable?

Consider the topology of radially open sets on $\mathbb{R^2}$ : a set $S$ is radially open iff, for every $x\in S$ and every line $L\subset\mathbb{R^2}$ that contains $x$, $S$ contains an open segment ...
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45 views

Proving a well known result of metric spaces without using Zorn's Lemma.

I saw a proof of the result using Zorn's Lemma: Statement: If $X$ is a metric space in which every uncountable set has a limit point,then $X$ is separable. The proof was a bit constructive and does ...
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1answer
40 views

About countable set

Let $f$ be a function in [a,b] with a countable break points (note by $S$). Is it true that the closure of S is countable?
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27 views

$Y\subset X$ is Lindelof iff Every $X$-open cover of $Y$ has a countable $X$-open subcover. [duplicate]

Suppose $(X,\tau)$ is a topological space and $A\subset X$,suppose $\{G_\alpha\}$ is an open cover of $A$ i.e. $A\subset \cup G_\alpha$,does there exist a countable subcover $\{G_{\alpha_n}\}$ of $A$....
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41 views

Separable vs countable

I want to show that $l^2(X)$ is separable iff $X$ is countable. Note that a space is separable if it has a countable dense subset. I can see that if $X$ is countable, then $l^2(X)$ is separable. To ...
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31 views

Is the set $A=\{x\in l_p :x_n \in \mathbb Q\}$ countable? [closed]

Is the set $A=\{x\in \it l_p :x_n \in \mathbb Q\}$ countable?If yes,then I can show that $l_p$ space is separable with resepect to the metric $d(x,y)=(\sum _nx_n^p)^{1/p}$.
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1answer
59 views

Every subspace of a separable metric space is separable

I'm trying to prove the following statement: Every subspace of a separable metric space is separable. Could you please verify if I correctly apply the concept relative topology? Thank you so much! $...
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1answer
32 views

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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1answer
49 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 ...
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2answers
40 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!
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23 views

Disjoint spaces and Countability

Let $(X_j)_{j\in J}$ be an indexed family of non-empty topological spaces. If $X_j$ second countable for each j and $\coprod_{j\in J}X_j$ is second countable. Show that $J$ is countable. Note: $\...
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1answer
36 views

Preservation of Second Countability

Let $f: X\rightarrow Y$ be a continuous open map. Show that if $X$ satisfies the second countability axiom, then so does $f(X)$. My attempt:Let $B_1,B_2,B_3....$ be an enumeration of the basis $\...
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29 views

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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1answer
41 views

Proof verification: Every separable metrizable space is second-countable.

I am aware that this question has been asked multiple times on this site; however none of them seem to have a proof as simple as mine. So I would like to know if this is legitimate: Let $A=\{x_n:n\in\...
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1answer
25 views

Secound Countable spaces and perfect maps

So im doing an exercise from Munkres, that is that if we have a perfect map $p : X \rightarrow Y$ then $X$ secound countable implies $Y$ secound countable. Im having some trouble seeing that the sets ...
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1answer
42 views

Generalization of G-delta Sets

A $G_\delta$-set is obtained from a countable intersection of open sets. Do we have any results about the intersection of uncountably many open sets? Or does it even make sense? The $G_\delta$-sets ...
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1answer
70 views

0-manifolds are countable and discrete

Want to show every 0 -manifold is a countable and discrete space. Let M be a 0-manifold. By definition, it is second countable, Hausdorff and each point in M has a neighborhood homeomorphic to $\...
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1answer
191 views

$\mathbb{R^n}$ 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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3answers
99 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:...
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1answer
84 views

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 ...
2
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3answers
90 views

$L^{p}$ 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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2answers
189 views

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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1answer
94 views

Second countable CW complexes

When exactly is a CW complex second-countable?
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112 views

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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1answer
41 views

In a locally compact 2nd-countable Hausdorff space $E$ there is a sequence of compact subsets $K_n$ with $K_{n-1}⊆\overset∘{K_n}$ and $\bigcup_nK_n=E$

Let $(E,\tau)$ be a locally compact second-countable Hausdorff space. I want to show that there is a $(K_n)_{n\in\mathbb N_0}\subseteq E$ such that $K_n$ is compact and $K_{n-1}\subseteq\overset{\circ}...
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1answer
65 views

Similarity and Difference between Separable Space and Separated space?

Does separability and/or second countability implies $T_2$ or higher axiom sets? My intuition is "no". Even $T_0$ space can be separability and/or second countability?
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33 views

Separable $\Rightarrow$ Lindelöf for metric spaces without using second-countability [duplicate]

It is well-known that for metric spaces, being separable, strongly Lindelöf and second-countable are equivalent. I know how to prove the equivalence between separable and second-countable, and I guess ...
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2answers
157 views

Is it true that the cardinality of the topology generated by a countable basis has at most cardinality $|P(\mathbb{N})|$?

I'm curious about what the cardinalities of second countable spaces can be at most. I have an idea as to how to show that the topology generated by a countable basis (which, for basis $B = \{B_i\}_{i=...
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1answer
248 views

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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2answers
244 views

Uncountable product of second countable spaces

Let $(X_i, \mathcal{T}_i)_{i \in I}$ be a family non indiscrete topological spaces and equip the product with the product topology $\mathcal{T}$. If $\prod X_i$ is second countable, prove that $|I|...
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1answer
42 views

$\coprod_{i \in I} X_i$ is second countable $\implies |I| \leq |\mathbb{N}|$

Let $(X_i \neq \emptyset)_{i \in I}$ be a family topological spaces. Prove that: $\coprod_{i \in I} X_i$ is second countable $\implies |I| \leq |\mathbb{N}|$ (the coproduct is equipped with the ...
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1answer
128 views

$\mathbb{R}_{\ell}$ is not second countable

For each $x\in \mathbb{R}_{\ell}$, there exist $r_1,r_2$ rational numbers such that $r_1<x<r_2$. Then take $\cal{B}=\{[r_1,r_2):r_1<x<r_2,x\in \mathbb{R}_{\ell}\}$. Then, for each $x\in\...
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0answers
30 views

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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2answers
217 views

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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1answer
303 views

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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1answer
69 views

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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2answers
313 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 ...
2
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2answers
170 views

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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2answers
91 views

Countable union covers a second countable, compact Hausdorff space

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 $...