Questions tagged [second-countable]
For questions about second-countable topological spaces, i.e., space with countable base.
178
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Prove that $S^2$ has a countable basis $\{U_n\}$ implies that $P^2$ has a countable basis $\{p(U_n)\}$.
The Problem: Let $p: S^2\to P^2$ be the quotient map from the $2$-sphere $S^2$ to its projective plane $P^2$. Then $S^2$ has a countable basis $\{U_n\}$ implies that $P^2$ has a countable basis $\{p(...
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How many second-countable $T_1$ spaces are there?
How many second-countable $T_1$ spaces, up to homeomorphism, are there?
Let $X$ be a second-countable $T_1$ space. Since $X$ is second-countable, there are at most $\beth_1$ open sets in $X$. And ...
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Are first and second countability preserved under intersection of topologies?
For a given set $X$ endowed with two topologies $\mathcal{T}$ and $\mathcal{T}'$, i.e. such that $(X,\mathcal{T})$ and $(X,\mathcal{T}')$ are two topological spaces defined on
the same $X$, it is easy ...
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Proof verifying that separable metric space is second countable
Show that a separable metric space $X$ is second countable.
I’m trying to figure out whether I got this proof correct.
Since $X$ is separable there exists a countable subset $D$ such that the ...
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Proof verification for separable metric space implies second countable.
Let $(X,d)$ be a separable metric space and let $Y \subset X$ be its countable dense subset.
Take the basis to be
$$\mathcal{B}=\{B_{\frac{1}{n}}(x): x \in Y, n \in \Bbb{Z}^+\}$$
I claim for an ...
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Second countability of $\mathbb{R}$
Claim:
$\mathbb{R}$ is second-countable, i.e. there exists a countable family of open sets such that any open subset of $\mathbb{R}$ can be written as countable union of a subfamily.
Proof:
Let $U$ be ...
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Let $X$ be a topological space that satisfies the second axiom of countability. Show that if ...
Let $X$ be a topological space that satisfies the second axiom of countability. Show
that if $\cal{B}$ is a basis for $X$ then there is $\cal{B'} \subseteq \cal{B}$ countable such that $\cal{B'} $ is ...
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Which axioms of countability does Michael's Line satisfy?
Which axioms of countability does Michael's Line satisfy?
Michael Line is the real line with the topology generated by $\cal{B}$ where $\cal{B}$ = {$\tau \cup ${{$x$}:$x \in \mathbb R$ \ $\mathbb Q$}...
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(Countability axioms)...What countability axioms does the topological space $(X, \tau )$ satisfy?
Given $X$ an infinite set and $x_0 \in X$, we know that
$B = ${{$x$}$ : x \in X$ and $x \ne x_0$} $\cup$ {$A \subseteq X : x_0 \in A$ and $X$ \ $ A$ is finite} is the basis for the topology
$\tau = ${$...
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Compact Hausdorff Space - X Second Countable iff C(X) separable
I recently stumbled across a property of compact Hausdorff spaces which is supposedly well-known, namely:
If $X$ is a compact Hausdorff space, then $X$ is second countable if and only if $C(X)$ is ...
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The Borel $\sigma$-algebra generated by the product topology coincides with the product of Borel $\sigma$-algebras: where did I get wrong?
Let $(\Omega_n, \tau_n)_n$ be a sequence of metrizable topological spaces. Let $\sigma (\tau_n)$ be the Borel $\sigma$-algebra on $\Omega_n$. Let $\Omega :=\prod_{n =1}^\infty \Omega_n$ and $\pi_n: \...
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Confusion about a proof of Lusin's theorem
I'm reading about Lusin's theorem in textbook Optimal Transport for Applied Mathematicians
Let us be more precise: take a topological space $X$ endowed with a finite regular measure $\mu$ (i.e. any ...
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Second countable $T_1$ space is sequentially compact if and only if it is compact
I need to show that a second countable $T_1$ space is sequentially compact if and only if it is compact. But I currently need only show that sequentially compact $\Rightarrow$ compact.
Suppose $X$ is ...
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Is the Alexandroff extension of a locally compact, second-countable space second-countable?
If $X$ is a locally compact, second-countable topological space, then is its Alexandroff extension $X^*$ also second-countable?
Our definition of locally compact is that for every $x$ in $X$, we have ...
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$(\Bbb R, \tau)$ with $\tau$ defined as: $U\in \tau \iff U=\emptyset \vee 0 \in U$ is not second-countable.
Let $(\Bbb R,\tau)$ be a topological space with $tau$ is defined as:
$$U\in \tau \iff U=\emptyset \vee 0 \in U.$$
Show that $\tau$ is not second-countable.
Attempt:
The goal is to show that for any ...
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Borel $\sigma$-algebra with $\sigma$-finite measure: does a function vanish almost everywhere outside its essential support?
Let $(X, \mathcal B(X), \mu)$ be a $\sigma$-finite measure space with $\mathcal B(X)$ the Borel $\sigma$-algebra of a topological space $X$. The essential support of $f:X \to \mathbb R$ is defined as
$...
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Why does satisfying the countability conditions make topological spaces so nice?
Motivation for the question:
While I can appreciate the significance of Theorem 30.1 below, and also the Urysohn metrization theorem, I can't seem to understand why either of them is rooted in these ...
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Concerning topological manifolds: Are paracompact and connected locally euclidian Hausdorff spaces always second-countable?
There are different definitions for topological manifolds, sometimes second-countability or paracompactness are added to being locally euclidian Hausdorff. (Sometimes Hausdorff is also left out, but I ...
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Chapter 11, Theorem 5.2 (4) of James Dugundji Topology
Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$.
Dugundji’s proof:
Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all ...
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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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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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