# Questions tagged [second-countable]

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

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### The countability of the Cantor-Bendixson rank in Polish spaces

I've been studying the Cantor-Bendixson theorem and have some questions about the proof of the countability of the Cantor-Bendixson rank in Polish spaces. I would greatly appreciate your insights on ...
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### Support of function

As reported in https://en.wikipedia.org/wiki/Support_(mathematics), for a function $f:X \rightarrow \mathbb{R}$ we can define some notion of $supp(f)$, in particular: If $X$ is only a set, we define ...
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### Does there exist a second-countable locally connected space with no countable basis of connected sets?

Space $X$ is called locally connected if it has a basis consisting of connected sets. It's called second-countable if it has a countable basis. If $X$ is both locally connected and second-countable, ...
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### Is every second countable locally compact Hausdorff space normal?

I know that there are locally compact Hausdorff spaces that aren't normal. Are there examples which are second countable?
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### Why is a Countable Basis Needed in This Proof?

My question is regarding the Theorem in Munkres that states: Every Regular Space with a Countable Basis is Normal. Before reading the proof in Munkres, I tried to prove it myself and came up with a &...
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### For non second-countable TVS, is the sum of measurable functions again measurable? [duplicate]

Let $(\Omega, \mathcal A)$ be a measurable space and $E$ a topological vector space. Let $f,g:\Omega \to E$ be measurable. I already proved that Theorem $E$ is second-countable, then $f+g$ is ...
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### Prove that the following properties are all finitely productive

The question goes as follows: Prove that the following properties are all finitely productive (1) $T_0$ and $T_1$ (2) Separable (3) First Countable (4) Second Countable (5) Finite (i.e., the ...
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### Locally compact 2nd countable Hausdorff space and complete metrizability

I was recently trying to verify certain things in the setting of Locally compact 2nd countable Hausdorff spaces. I thought that this is a natural collection of spaces more general than metric spaces, ...
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### How many second-countable $T_1$ spaces are there? [duplicate]

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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### 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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### 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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### 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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### 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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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 ...