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Questions tagged [second-countable]

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

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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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Theory & problems of set theory & related topics [closed]

Let S be the set of rational points in the plane R². Show that S can be partitioned into two sets V and H such that the intersection of V with any vertical line is finite and the intersection of H ...
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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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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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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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174 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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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 $|I|...
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$\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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70 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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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 ...
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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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49 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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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 ...
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108 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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68 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 $...
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Unit ball of the adjoint space of a separable Banach space is second-countable in the weak* topology.

In the proof : If $A$ is separable, then $\Delta(A)$ satisfies the second axiom of countability. ($\Delta(A)$ the set of all complex homomorphisms of $A$.) I have found they says "It is easy to see ...
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1answer
193 views

Continuous function and first and second countable spaces

I have 2 questions related to the proof given here, In I, why to propose $\{F(B_i) \cap F(X)\}_{i \in I}$ as a basis for $f(p)$ when $\{B_i\}_{i \in I}$ is a basis for $p$? That is what we want to ...
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Product topology and countable topology spaces

Let $I\neq\emptyset$ numerable and $(X_\alpha,\tau_\alpha)$ a family of topological spaces.Prove the following. $\displaystyle\prod X_\alpha$ is first-countable if and only if $X_\alpha$ is ...
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Space of holomorphic functions with compact-open topology is second-countable

Basically what the title says. Can anybody provide a proof that the space of holomorphic functions on an open subset $U \subset \mathbb{C}$ equipped with the compact-open topology (a.k.a. topology of ...
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$X^*$ with $w^*$-topology second countable for separable Banach space $X$?

I'm wondering if it is generally true that for any separable Banach space $X$ the space of linear, continuous functionals $X^*$ equipped with the $w^*$-topology is second countable. If so, how can I ...
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How does the underlying topological space of manifold depend on the properties of the atlas?

Let $X$ be a set. In Serge Lang's "Differential and Riemannian manifolds" the following definition of smooth $\mathbb{R}^{n}$-atlas on $X$ is given. Let $I$ be some index set. An atlas of class $C^...
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247 views

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 ...
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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 ...
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1answer
137 views

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 ...
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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 ...
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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 ...
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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 ...
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1answer
128 views

$X$ 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 ...
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1answer
58 views

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 ...
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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,...
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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 ...
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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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If X is second-countable, then X is Lindelöf.

Munkres in his book states that: Theorem 30.3 Suppose that $X$ has countable basis, then every open covering of $X$ contains a countable subcollection covering $X$. $\textbf{Proof.}$ Let $...
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159 views

Show that any countable subset of the reals is negligible

I got this task : Show that any countable subset of the reals is negligible. Does this mean that the subset of the reals has an outer zero measure? I'm quite new too measure theory, so dont know quit ...
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Why are manifolds second countable [duplicate]

A manifold is defined to be a second countable Hausdorff space that is locally homeomorphic to Euclidean space. It surprises me that second countably is a requirement here, because it surprises me ...
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453 views

An uncountable set in a second countable space has limit points.

The problem I tried to solve is the following: Let $(X,\tau)$ a second countable topological space. Let $A\subset X$ such that $\text{card}(A)>\mathbb{N}$. Then $A$ has accumulation points. ...
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Second Countability Preserved under CLOSED Continuous Surjection

In my General Topology course we were recently shown the theorem that says that Second Countability Preserved under Open Continuous Surjection and the natural question is if we can switch the "Open" ...
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1answer
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second-countability of the upper halfplane

How to prove that the upper half plane, including the line $\{y=0\}$ with the subspace topology derived from the topology induced by euclidian open bols in $\mathbb{R}^2$ is separable and first-...
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1answer
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If $(E,\tau)$ is a topological space $E_2$, then $(E,\tau)$ is Lindelöf and separable.

I can't prove this statement: If $(E,\tau)$ is a topological space $E_2$ (There is a countable basis for $\tau$), then $(E,\tau)$ is Lindelöf and separable. I tried to prove Lindelöf first. Let $C = ...
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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 ...
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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 ...
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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. ...
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1answer
163 views

Does every compact metric space have a finite basis?

Let $M$ be a compact metric space. Then the collection $\mathscr{U}=\{B(x,\frac1n):x\in M, n\in\Bbb{N}\}$ forms an open cover of $M.$ Let $\mathscr{B}_0$ be a finite sub-cover extract from $\mathscr{U}...
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How can I prove 'separable' guarantees 'second countable'?

Separable means having a countable dense set and second countable means having a countable basis. But I cannot find any relation between dense set and basis.
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cardinality of the Borel $\sigma$-algebra of a second countable space

Second countability by itself doesn't restrict the cardinality of a topological space, since every set with the trivial topology is a second countable space, but it seems natural to ask whether second ...
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Constraining a dense sequence on a product space, one factor at a time

Slogan: Given a sequence on $X\times Y$, can we choose subsequences to fix the limit in $X$ while leaving the behavior on $Y$ free? Details: Suppose $X$ and $Y$ are topological spaces and $(x_n,y_n)...
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383 views

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 ...
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3answers
262 views

How to show that $\omega_1$ is not secound countable?

I wish to produce a direct proof that $\omega_1$ is not second countable Recall the definition of $\omega_1$ $\omega_1 = \{\alpha \in W| \{x \in W| x < \alpha\} \text{ is countable }, W \text{ is ...
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1answer
271 views

Every compact metrizable space is second countable

I am trying to show Every compact metrizable space is second countable My Attempt: Let $(X,\mathfrak{T})$ be a compact metrizable space. We wish to show that it has a countable basis. Then given $C ...