All Questions
Tagged with second-countable manifolds
7 questions
3
votes
1
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388
views
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 ...
- 2,184
3
votes
2
answers
226
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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.
...
- 3,599
1
vote
1
answer
875
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Manifold has a countable cover by compact sets.
I was confused by Boothby's proof that a Manifold had a countable cover of compact sets. In his proof he says that as $M$ has a countable basis of open sets, we may assume that $M$ has a countable ...
- 2,833
1
vote
1
answer
440
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0-manifolds are countable and discrete
Want to show every 0 -manifold is a countable and discrete space.
Let M be a 0-manifold. By definition, it is second countable, Hausdorff and each point in M has a neighborhood homeomorphic to $\...
user643073
0
votes
1
answer
4k
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$\mathbb{R}^n$ Second Countable
Lemma: Let $U$ be an open subset of $\mathbb{R}^n$ and $x \in U$ then there exists a ball with rational center $(\mathbb{Q}^n$ and rational radius containing $x,$ that is contained in $U.$
What I want ...
user643073
3
votes
2
answers
679
views
Does partition of unity implies second countable?
Reading the definition of partition of unity:
Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\Phi$ of $C^\infty$ functions $\varphi$ defined in an ...
user97512
3
votes
0
answers
116
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Why are manifolds second countable [duplicate]
A manifold is defined to be a second countable Hausdorff space that is locally homeomorphic to Euclidean space.
It surprises me that second countably is a requirement here, because it surprises me ...
- 31.3k