Questions tagged [second-countable]

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

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Exercise 4, Section 30 of Munkres’ Topology

Show that every compact metrizable space $X$ has a countable basis. [Hint: Let $\mathscr{A}_n$ be a finite covering of $X$ by $1/n$-balls.] My attempt: Approach(1): $B_n =\{ B(x, \frac{1}{n})| x\in X\...
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Exercise 5, Section 30 of Munkres’ Topology

(a) Show that every metrizable space with a countable dense subset has a countable basis. (b) Show that every metrizable Lindelof space has a countable basis. My attempt: (a) Since $X$ is separable, $...
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Definition of Countability in Munkres’ Topology

$X$ have a countable basis at $x$, if $\exists \{U_n \in \mathcal{N}_x |n\in \Bbb{N}\}$ with the following property: $\forall U\in \mathcal{N}_x, \exists m\in \Bbb{N}$ such that $U_m\subseteq U$. If $...
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Theorem 30.3 of Munkres’ Topology

Suppose that $X$ has a countable basis. Then: (a) Every open covering of $X$ contains a countable subcollection covering $X$. (b) There exists a countable subset of $X$ that is dense in $X$. ...
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Prob. 5 (b), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizable Lindelof space is second-countable

Here is Prob. 5 (b), Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that every metrizable Lindelof space has a countable basis. My Attempt: Let $X$ be a metrizable Lindelof ...
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Prob. 5 (a), Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Every metrizabe separable space is second-countable

Here is Prob. 5 (a), Sec. 30, in the book Topology by James R. Munkres, 2nd edition: Show that every metrizable space with a countable dense subset has a countable basis. My Attempt: Let $X$ be a ...
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On second-countable space and its uncountable subset

This is Munkres Chapter 4, Section 30, Problem Number 3. Let $X$ have a countable basis; let $A$ be an uncountable subset of $X$. Show that uncountably many points of $A$ are limit points of $A$. My ...
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Topology counterexamples without ordinals

I am looking for three counterexamples in general topology, namely: A set which is sequentially closed, but not closed; A set which is sequentially compact, but not compact; A set which is compact ...
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Proof that Metric Spaces are Second Countable?

Metric spaces can be equipped with the topology given by the open sets (which in turn are defined with the help of open balls) such that metric spaces are topological spaces. Now, the definition of ...
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Let $(E,d)$ be a metric space. Then $X$ is second-countable if and only if $X$ is Lindelöf if and only if $X$ is separable

In proving every subspace of a separable metric space is separable, I need below result. Could you check if my proof is fine? Theorem: Let $(E,d)$ be a metric space. Then $X$ is second-countable if ...
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$f:I\rightarrow X$ where $X$ is hausdorff show that $X$ is metrizable.

This question comes from section 44 problem 4 of Munkres. Let $X$ be a Hausdorff space. Let $I=[0,1]$. Show that if there is a continuous surjective map $f : I \rightarrow X$, then $X$ is compact, ...
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Example of a topological space that has countably (but not finitely) many elements, is regular, but not 2nd countable

I'm trying to find an example of a topological space that has countably (but not finitely) many elements, is regular, but not 2nd countable. It seems to me that if a space has countably many elements, ...
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Chain of compact sets in locally compact space

When a topological Hausdorff space X is locally compact and second-countable (has countable weight), can we find a chain of compact sets $\{K_i: n \in \mathbb{N}\}$, where $K_0 = \emptyset$ and $K_n \...
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How to tell if a space is second-countable

A topological space is called second-countable iff it has a countable basis. How to prove or at least make an assumption about whether a space does, or does NOT have countable basis? Which properties ...
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Does every Lie group have at most countably many connectected components?

Some proofs in a lecture I took were motivated by this statement that "some people don't assume second countability when they define a topological manifold, but for Lie groups we get this ...
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Proving that a space is second countable

Let $(X, \tau_X)$ compact, $(Y, \tau_Y)$ $T_2$ and $f:(X, \tau_X) \to (Y, \tau_Y)$ a continuous and surjective function. Prove if $(X, \tau_X)$ is second countable then $(Y, \tau_Y)$ is second ...
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Example of a Hausdorff topology that is homeomorphic to $\mathbb R^n$ but isn't second countable?

The definition of a topological manifold $M$ I have is: $M$ is Hausdorff. Each point of $M$ has a neighborhood that is homeomorphic to an open subset of $\mathbb{R}^n$. $M$ is second countable. ...
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Why the the new basis is countable?

I was thinking about the following question: If $p: X \rightarrow Y $ a continuous, closed, and surjective map with the property that for each $y \in Y$ we have $p^{-1}(\{y\})$ is compact. Prove that ...
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Does second countablity implies closed (or open) set in $X$ is a countable intersection of open sets in $X$?

Is $X$ is second countable implies or equivalent to Every closed(or open) set in $X$ is a countable intersection of open sets in $X$? I know, metric space implies this by Closed set as a countable ...
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On showing that every separable metric space has a countable base

Theorem: Every separable metric space $(M, d)$ has a countable base I've glanced over a couple of proofs for this theorem (for example see the one provided here), but I'm still stuck at the following:...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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1 answer
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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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1 answer
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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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4 votes
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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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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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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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1 answer
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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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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 $\mathbb{R}_\ell$. Recall ...
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1 answer
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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 of ...
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1 vote
1 answer
184 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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3 votes
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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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3 votes
1 answer
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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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