Questions tagged [second-countable]
For questions about second-countable topological spaces, i.e., space with countable base.
213 questions
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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 ...
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2
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134
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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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93
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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
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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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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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1
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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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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 ...
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1
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0
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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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0
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194
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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-...
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1
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92
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Basis of a lower limit topology
I’m trying to determine whether the lower limit topology (LLT) on $\mathbb{R}$ is second countable or not. I considered the set
$$
\mathcal{B} = \{ [a, b) \mid a, b \in \mathbb{Q} \}
$$
as a candidate ...
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2
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305
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Is a Hausdorff subtopology of a second countable space second countable? [closed]
Let $(X, \tau_1)$ be a Hausdorff second countable topological space and $(X, \tau_2)$ be a space with a coarser topology, but still Hausdorff. Is $\tau_2$ still second countable?
More in general, in ...
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92
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Mutually disjoint dense subsets in $\mathbb{R}$ [duplicate]
There exist three mutually disjoint subsets of $\mathbb{R}$, each of which is countable and dense in $\mathbb{R}$
For each $n \in \mathbb{N}$, there exist $n$ mutually disjoint subsets of $\mathbb{R}$ ...
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3
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1
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242
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A space with a sequence of metrizable subspaces is metrizable.
A compact Hausdorff space $X$ has a sequence of subspaces $A_1\subset A_2 \subset A_3 \subset \cdots $.
And $A_n$ has the following properties
・$\cup A_n =X$
・All $A_n$ are metrizable (as a relative ...
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3
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Right closed-ray topology on $\mathbb{R}$ is not second countable
I am taking the collection $\mathcal B$ of $\{ [a,\infty), a\in\mathbb{R}\}$, then this collection makes basis set on $\mathbb{R}$.
Now i am claiming that this not a second countable topology.
$\text{...
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