All Questions
Tagged with second-countable connectedness
6 questions
4
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2
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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, ...
- 11.9k
1
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2
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$f:I\rightarrow X$ where $X$ is hausdorff show that $X$ is metrizable.
This question comes from section 44 problem 4 of Munkres.
Let $X$ be a Hausdorff space. Let $I=[0,1]$. Show that if there is a continuous surjective map $f : I \rightarrow X$, then $X$ is compact, ...
- 346
4
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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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Topological properties of Sorgenfrey line.
Consider $\mathbb R$ with lower limit topology (generated by taking $[a,b)$ intervals as basis).The topological space thus generated is called Sorgenfrey line.What are some interesting properties of ...
- 3,801
7
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Countable basis but uncountably many connected components
Looking for some guidance on two topology questions:
(a) Show that a locally connected space with a countable basis, has at most
countably many connected components.
(b) Give an example when X has ...
3
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2
answers
2k
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Does there exist a Connected Locally Euclidean Space that is not second countable?
A problem in Lee's Introduction to Topological Manifolds got me thinking about this question. I can easily construct a locally euclidean space that is not second countable, by taking a disjoint union ...
- 15.6k