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Let $X$ be a nontrivial topological space. Assume ZFC.

If $X$ is a first countable compact $T_2$ space, then $|X|\leq c$.

If $X$ is a first countable separable $T_2$ space, then $|X|\leq c$.

Since second countability implies first countability and separability, I am guessing that:

If $X$ is a second countable $T_2$ space, then $|X|\leq c$.

My question:

  1. What is the cardinality of a connected separable $T_2$ space?
  2. What is the cardinality of a second countable $T_1$ space?
High GPA
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  • In your second line, did you mean $T1$ instead of $T2$? – Chickenmancer May 14 '22 at 16:48
  • @Chickenmancer Yes – High GPA May 14 '22 at 17:10
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    The Stone-Cech compactification of $\mathbb R$ has cardinality $2^c$, and it's connected and separable because $\mathbb R$ is. – Andreas Blass May 14 '22 at 17:51
  • And @AndreasBlass' observation is optimal, by a special case of the argument presented here. – Noah Schweber May 14 '22 at 17:53
  • @AndreasBlass So it seems like the cardinality of connected separable T2 space is less than or equal to $2^c$? – High GPA May 25 '22 at 19:33
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    @HighGPA The cardinality of any separable Hausdorff space (connected or not) is bounded by $2^c$. More generally, a Hausdorff space with a dense subset of size $\kappa$ cannot be larger than $2^{2^\kappa}$. The reason is that every point is the limit of a different filter on the dense set, and there are only $2^{2^\kappa}$ filters on a set of size $\kappa$. – Andreas Blass May 25 '22 at 19:40
  • @AndreasBlass So the property of "connected" does not usually impose a restriction on cardinality? – High GPA May 25 '22 at 21:06
  • @HighGPA That depends a lot on "usually". For example, normal ( = T1 + T4) spaces can have any cardinality at all, but connected ones can't have cardinalities strictly between $1$ and $c$. So in that situation, connectedness definitely affects cardinality. – Andreas Blass May 25 '22 at 21:30
  • There are a few countable, connected, Hausdorff ($T_{2}$) spaces listed on the $\pi$-base

    https://topology.jdabbs.com/spaces?q=countable%2Bconnected%2Bhausdorff

     – Robert Thingum May 26 '22 at 09:55

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