Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [second-countable]

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

A topological spaces $X$ is called second-countable if it has a countable base. This can be equivalently formulated as: $X$ has countable weight, i.e., $w(X)\le\aleph_0$.

200 questions
37
votes
1 answer

When is $C_0(X)$ separable?

Recall that a compact Hausdorff space is second countable if and only if the Banach space $C(X)$ of continuous functions on $X$ is separable. I'm looking for a similar criterion for locally compact Hausdorff spaces, using $C_0(X)$ (the Banach space…
Paul Siegel
  • 9,467
22
votes
2 answers

A metric space is separable iff it is second countable

How do I prove that a metric space is separable iff it is second countable?
Heisenberg
  • 3,387
21
votes
2 answers

Every separable metric space has a countable base

A collection $\{V_{\alpha}\}$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x\in X$ and every open set $G\subset X$ such that $x\in G$, we have $x \in V_\alpha \subset G$ for some $\alpha$. In other words,…
RFZ
  • 17,648
  • 11
  • 51
  • 144
18
votes
2 answers

Second-countable implies separable/Axiom countable choice

Let $(X,\mathscr T)$ be a topological space, and $(B_n)_{n\ge1}$ a countable basis for X. Under this assumptions, X is separable. The proof of this assertion is as follows: We can assume without loss of generality that all the $B_n$ are…
ditlew
  • 221
18
votes
4 answers

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 ${B_n}$ be a countable basis and $\mathcal{A}$ an open cover…
George
  • 3,957
17
votes
5 answers

Show that every compact metrizable space has a countable basis

Show that every compact metrizable space has a countable basis. My try: Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in X\}$ of $X$ .As $X$ is compact we can find finite…
Learnmore
  • 31,675
  • 10
  • 110
  • 250
16
votes
1 answer

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by neighborhoods of radius $\frac{1}{n}$, and we have a…
PandaMan
  • 3,337
16
votes
1 answer

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 $(x_n,y_n)_n$ is a sequence that is "limit-dense" in $X\times…
Chris Culter
  • 27,415
15
votes
2 answers

Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?

Background: This question came up in my homework (but was not a homework problem). The problem was proving one direction of the Heine-Borel theorem. As with all proofs of compactness, one begins with, "Suppose $A$ is closed and bounded, and…
Neal
  • 34,012
15
votes
2 answers

Every locally compact, second countable Hausdorff space has a countable basis of open sets with compact closure

Let $X$ be a locally compact, second countable Hausdorff space. I want to prove that it has a countable basis of opens with compact closure, and that this basis can be extracted as a subset of any basis of its topology by restricting to opens with…
ubugnu
  • 425
14
votes
1 answer

Hausdorff locally compact and second countable is $\sigma$-compact

I want to show that every Hausdorff, locally compact and second countable is $\sigma$-compact. I'm having trouble writing this rigorously. Can we proceed as follows: Let $X$ be a locally compact Hausdorff space. Let $\mathcal{B}=\{B_{n}: n \in…
user10
  • 5,806
14
votes
2 answers

When is the quotient space of a second countable space second countable?

I am a bit confused about this concept because I have read that the quotient space is second countable if the quotient map is open. However, I thought the definition of a quotient map was a surjective, continuous, open mapping. Suppose that $X$ is…
user190570
  • 232
13
votes
1 answer

Compact metric spaces is second countable and axiom of countable choice

Why we need axiom of countable choice to prove following theorem: every compact metric spaces is second countable? In which step it's "hidden"? Thank you for any help.
user63123
  • 195
12
votes
2 answers

Are countable topological spaces second-countable?

Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other hand both discrete and anti-discrete (or more…
freakish
  • 47,446
12
votes
1 answer

Is one-point compactification of a space metrizable

Let $X$ be a locally compact Hausdorff space.Let $Y$ be the one-point compactification of $X$. Two questions are: Is it true that if $X$ has a countable basis then $Y$ is metrizable? Is it true that if $Y$ is metrizable then $X$ has a countable…
Learnmore
  • 31,675
  • 10
  • 110
  • 250
1
2 3
…
13 14