Questions tagged [second-countable]
For questions about second-countable topological spaces, i.e., space with countable base.
213 questions
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Prob. 12, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is a continuous image of a first- / second-countable space is first- / second-countable?
Here is Prob. 12, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Let $f \colon X \longrightarrow Y$ be a continuous open map. Show that if $X$ satisfies the first or the second ...
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0
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68
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Prob. 17, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is the subspace $\mathbb{Q}^\infty$ of $\mathbb{R}^\omega$ in the box topology Lindelof? separable?
Here is Prob. 17, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Give $\mathbb{R}^\omega$ the box topology. Let $\mathbb{Q}^\infty$ denote the subspace consisting of sequences of ...
- 28.6k
2
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1
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224
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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 $f \colon X \longrightarrow Y$ be a continuous open map. Show that if $X$ satisfies the first or the second ...
- 28.6k
1
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0
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82
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Prob. 7, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is $\overline{S_\Omega}$ first-countable? second-countable? separable? Lindelof?
Here is Prob. 7, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Which of our four countability axioms does $S_\Omega$ satisfy? What about $\overline{S_\Omega}$?
Our four ...
- 28.6k
1
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0
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Prob. 7, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is $S_{\Omega}$ first-countable? second-countable? separable? Lindelof? [closed]
Here is Prob. 7, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Which of our four countability axioms does $S_{\Omega}$ satisfy? ...
Here the four countability axioms are (i) first-...
- 28.6k
0
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1
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304
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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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Proof that $\mathbf{Man}$ doesn't have coproducts [duplicate]
A smooth manifold is a locally Euclidean Hausdorff space which is second countable. The latter condition means that the category $\mathbf{Man}$ of smooth manifolds has countable coproducts, but it ...
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Prob. 8, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: Is this metric space first-countable? second-countable? separable? Lindelof? [closed]
Here is Prob. 8, Sec. 30, in the book Topology by James R. Munkres, 2nd edition:
Which of our four countability axioms does $\mathbb{R}^\omega$ in the uniform topology satisfy?
Here $\mathbb{R}^\...
- 28.6k
0
votes
1
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57
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Proving Lee's Thm 4.77 (Every second countable, locally compact Hausdorff space is paracompact) using weaker exhaustion lemma
The goal
My goal is to break down the proof of Theorem 4.77 from Introduction to Topological Manifolds by Lee which (as the title suggests) is:
Theorem 4.77. Let $X$ be a second countable, locally ...
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5
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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 ...
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1
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Given a connected Lie group $G$ why must its universal cover be a second countable space?
At the moment I am taking a first course in Lie groups and am using these notes. I am trying to prove the following proposition (proposition $3.5$ in the notes, page $24$), which I formulated ...
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Radially Open Topology on $\mathbb{R}^2$: Hausdorff, Countability, and Continuity [closed]
This problem appeared in (MATH GRE PREP: WEEK 2) form UCHICAGO 2019.
A subset $U \subseteq \mathbb{R}^2$ is radially open if for every $x \in U$ and every $v \in \mathbb{R}^2$, there exists $\...
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Investigating Countability Axioms for the Space of Ordinals $[0,\Omega)$, where $\Omega$ is the first uncountable ordinal.
I wish to discuss about the following question from general topology, involving set of ordinals:
Problem:
Let $X=[0,\Omega)$ be the set of all ordinals strictly smaller than the first uncountable ...
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1
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257
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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 ...
- 28.6k
3
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1
answer
370
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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 ...
- 28.6k