All Questions
Tagged with second-countable examples-counterexamples
12 questions
2
votes
1
answer
84
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Find an example of connected topology for finite space $\{1,...,n\}$ that conforms with the intuition of a commoner.
The most common topology for finite space is the discrete topology, which is clearly not connected.
There are some other discussion on this site, which give many example topology of finite space and $\...
- 862
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
5
votes
3
answers
3k
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Compactness gives Second countable space??
Suppose that $(X,\mathcal{T})$ is a topological space. If we know that $X$ is compact can we assume that is also second countable ??
Because X is compact we have that $X=\cup_{i=1}^{n}V_{i}$, where $...
- 1,103
1
vote
1
answer
290
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Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has no countable dense subset [relative to the subspace topology].
Let $\mathbb R_l$ be $\mathbb R$ given the lower limit topology, i.e. the topology generated by $\{[a, b) \subseteq \mathbb R|a<b\}$.
Give a subspace of $\mathbb R_l \times \mathbb R_l$ that has ...
- 269
6
votes
2
answers
3k
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First countable + separable imply second countable? [closed]
In topological space, does first countable+ separable imply second countable? If not, any counterexample?
- 7,744
6
votes
2
answers
2k
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Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness?
The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something?
...
14
votes
2
answers
6k
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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,...
- 232
2
votes
1
answer
634
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If a topological space $S$ is second-countable, must necessarily every quotient space of $S$ be second-countable?
Let $S$ be a second countable topological space. Let $S^*$ be a quotient space of $S$ with quotient map $\pi$. If $\pi$ is open, it's easy to show that it transfers a basis of $S$ into a basis of $S^*$...
- 5,311
7
votes
2
answers
2k
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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 ...
11
votes
2
answers
2k
views
Compact topological space not having Countable Basis?
Does there exist a compact topological space not having countable basis?
I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. Tychonoff'...
- 1,057
5
votes
2
answers
2k
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locally compact Hausdorff space which is not second-countable
I'm trying to find an example of a space that is Hausdorff and locally compact that is not second countable, but I'm stuck. I search an example on the book Counterexamples in Topology, but I can't ...
- 1,346
3
votes
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