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 ...
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59 views

Why is the topology on the Sorgenfrey line not second countable?

For context, let me clarify some things. Our set is $\mathbb{R}$. The topology on $\mathbb{R}$ is the topology generated by the arbitrary unions of closed-open intervals $[a,b)$ with $a,b \in \mathbb{...
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44 views

First countable, second countable, and Lindelöf

I've studying the countability chapter in Schaum's General Topology and in the book, the author goes over first countable, second countable, and Lindelöf spaces. Once the author covers the definitions ...
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38 views

Why is this space not second-countable?

This is a problem in Lee's Introduction to Smooth Manifolds. Show that a disjoint union of uncountably many copies of $\mathbb{R}$ is not second-countable. Let $S$ be the disjoint union of ...
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44 views

Topological properties of Sorgenfrey line.

Consider $\mathbb R$ with lower limit topology (generated by taking $[a,b)$ intervals as basis).The topological space thus generated is called Sorgenfrey line.What are some interesting properties of ...
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13 views

Has the topology of inferior semicontinuity a countable base?

Let $X=\mathbb{R}$ and $\tau$ the inferior semicontinuity topology, defined as: $$\tau=\{(a,+\infty) \mid a \in \mathbb{R}\}\cup\{\mathbb{R}\}$$ where $(a,+\infty)=]a,+\infty[$ is an open interval. ...
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1answer
40 views

A second countable space is countably compact iff it is compact

Compact implies countably compact. But I'm having difficulty with proving the other direciton. My work: Suppose $(X, \mathcal{O})$ is second countable and countably compact. By definition, there is a ...
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38 views

For what definition of local compactness are locally compact and second-countable spaces $\sigma$-compact?

We know that a Hausdorff, secound-countable, and locally compact space is sigma-compact. As I understand, however, the Hausdorff requirement is typically attached to avoid choosing a particular ...
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18 views

Second countability from a continuous open map

Exercise 1.3.10 of Naber's "Topology, Geometry and Gauge fields - Foundations (2nd Edition)" reads Show that if $f:X \rightarrow Y$ is a continuous, open mapping from $X$ onto $Y$ and $X$ ...
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49 views

Help me visualize a Lindelöf space and a corresponding theorem

I am self studying Metric Spaces from the book by Satish Shirali and H.L. Vasudeva. I am having trouble in visualizing how a Lindelöf space looks like. The book defines it as: A metric space is said ...
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34 views

Is the countable union of second countable open subspaces second countable?

I need help checking if the statement below is true. I have written a proof to it. If the statement is incorrect I'd greatly appreciate it you could point out where I went wrong in my proof. Thank you....
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51 views

Counter example in topology [closed]

We known that every subspace of second countable space is second countable. I was think if A is subspace of X and A is second countable space need to be the whole space is second countable space? My ...
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52 views

Does second countable implies countably many components?

I was proving that a manifold has countably many connected components and this question came up. The way I did the original question is to show that components in manifolds are open and then use the ...
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30 views

Second countable space $\implies [x_n\to a, f(x_n)\to L \implies \lim\limits_{x\to a}f(x)=L]$

Let $f:S\to Y$ where $S\subset X$. I wanted to know if $\lim\limits_{x\to a}f(x) = L$ (where $a$ is a limit point of $A$) was equivalent to $\lim\limits_{n\to \infty}f(x_n) = L$ for all sequences $x_n\...
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42 views

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 ...
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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 ...
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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. ...
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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?
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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 ...
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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 ...
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108 views

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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62 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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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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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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35 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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97 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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68 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
86 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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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
128 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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52 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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41 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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$Y\subset X$ 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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46 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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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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Second Countability $\implies$ Lindelöf property(Proof verification)

I am a beginner in metric space course. Recently I have learnt the terms second countability and Lindelöfness of a space(metric/topological).Now I have proved that second countability implies ...
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1answer
106 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
76 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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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 ...
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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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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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48 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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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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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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3answers
825 views

Lindelöf and second countable spaces

Can anyone give me some examples and non examples of Lindelöf or second countable space and spaces that is Lindelöf but not second countable? And I understand the definition but find it is hard to ...
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58 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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76 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
115 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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1k 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
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:...