# Questions tagged [second-countable]

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

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### Compact Hausdorff Space - X Second Countable iff C(X) separable

I recently stumbled across a property of compact Hausdorff spaces which is supposedly well-known, namely: If $X$ is a compact Hausdorff space, then $X$ is second countable if and only if $C(X)$ is ...
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### Why does satisfying the countability conditions make topological spaces so nice?

Motivation for the question: While I can appreciate the significance of Theorem 30.1 below, and also the Urysohn metrization theorem, I can't seem to understand why either of them is rooted in these ...
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### 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 ...
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### Chapter 11, Theorem 5.2 (4) of James Dugundji Topology

Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$. Dugundji’s proof: Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all ...
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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 ...
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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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### 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 ...
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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:...
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### 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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### Help me visualize a Lindelöf space and a corresponding theorem

I am self studying Metric Spaces from the book by Satish Shirali and H.L. Vasudeva. I am having trouble in visualizing how a Lindelöf space looks like. The book defines it as: A metric space is said ...
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