# Questions tagged [second-countable]

For questions about second-countable topological spaces, i.e., space with countable base.

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### How to tell if a space is second-countable

A topological space is called second-countable iff it has a countable basis. How to prove or at least make an assumption about whether a space does, or does NOT have countable basis? Which properties ...
122 views

### 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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### Proving that a space is second countable

Let $(X, \tau_X)$ compact, $(Y, \tau_Y)$ $T_2$ and $f:(X, \tau_X) \to (Y, \tau_Y)$ a continuous and surjective function. Prove if $(X, \tau_X)$ is second countable then $(Y, \tau_Y)$ is second ...
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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. ...
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### Why the the new basis is countable?

I was thinking about the following question: If $p: X \rightarrow Y$ a continuous, closed, and surjective map with the property that for each $y \in Y$ we have $p^{-1}(\{y\})$ is compact. Prove that ...
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### Does second countablity implies closed (or open) set in $X$ is a countable intersection of open sets in $X$?

Is $X$ is second countable implies or equivalent to Every closed(or open) set in $X$ is a countable intersection of open sets in $X$? I know, metric space implies this by Closed set as a countable ...
1 vote
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### On showing that every separable metric space has a countable base

Theorem: Every separable metric space $(M, d)$ has a countable base I've glanced over a couple of proofs for this theorem (for example see the one provided here), but I'm still stuck at the following:...
60 views

### Second Countability Implies Separability

Munkres states explicitly that: Theorem 30.3: Suppose that $X$ has a countable basis. Then: (b) There exists a countable subset of $X$ that is dense in $X$. The following proof he gives is ...
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### Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal?

Let X be a topological space. a) If X is Hausdorff and it has an countable base, so X is normal? b) If X is regular, and if each point of X has a fundamental countable neighborhood system and is ...
1 vote
43 views

### Is this topology separable, first countable, second countable?

Consider a set $E$ with some point $p \in E$ and the topology: $T = \left \{ \emptyset \right \}\cup \left \{ C \subset E: p \in E \right \}$. Is this topology, separable, first countable or second ...
66 views

### Examples of topologies in which all open sets are countable union of base elements, but the topology itself doesn't have countable bases.

In second countable topology(i.e. has countable bases), obviously all open sets are countable union of base elements. But the inverse seems not true. The sorgenfrey line $R_l$ seems to be an example. ...
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### Second countable space implies numerable subbasis [closed]

If a topological space (X,t) has a numerable basis, implies X has a numerable subbasis?
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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 ...
Let $X, Y$ be topological spaces and $C(X,Y)$ the set of continuous functions with the compact open topology defined as having the subbasis consisting of all subsets $C(K, U)$ of functions $f$ for ...