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Let $(X,\tau)$ be a topological space and $Y\subset X$ be such that $|Y|=|\mathbb{R}|$ and $(Y,\tau|_y)$ is a discrete space. Does it follow that $(X,\tau)$ is not separable?

What about the reciprocal, if $X$ is not separable can we always find a discrete uncountable subspace?

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    $\begingroup$ I found an answer to my main question here. $\endgroup$ – Anguepa Feb 17 '16 at 16:55
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Let $\Psi$ be maximal family of almost disjoint subsets of $\mathbb{N}$ that has cardinality continuu. Consider the associated Mrówka space $X_\Psi = \mathbb{N} \cup \Psi$. Then $\Psi$ is a discrete subspace and $X_\Psi$ is separable because $\mathbb{N}$ is dense in it.

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The answer to your secondary question is also no. One counterexample is $\mathscr{F}[\Bbb R]$, the Pixley-Roy hyperspace of the reals. As a set, $\mathscr{F}[\Bbb R]$ is simply the family of all non-empty finite subsets of $\Bbb R$. For each $F\in\mathscr{F}[\Bbb R]$ and each open nbhd $U$ of $F$ in $\Bbb R$ we define

$$[F,U]=\{G\in\mathscr{F}[\Bbb R]:F\subseteq G\subseteq U\}\;;$$

the collection of all such sets $[F,U]$ turns out to be a base for a topology on $\mathscr{F}[\Bbb R]$, called the Pixley-Roy topology. It’s a rather nice topology: it makes $\mathscr{F}[\Bbb R]$ a zero-dimensional Hausdorff space, hence Tikhonov. All of this is verified in the linked article, as are the following observations:

  • Since $\Bbb R$ is uncountable, $\mathscr{F}[\Bbb R]$ is not separable. (Point $6$ in the article.)
  • Since $\Bbb R$ is second countable, $\mathscr{F}[\Bbb R]$ does not contain an uncountable family of pairwise disjoint open sets. (The technical term is that $\mathscr{F}[\Bbb R]$ has countable cellularity.) (Point $14$.)
  • Since $\mathscr{F}[\Bbb R]$ does not contain an uncountable family of pairwise disjoint open sets, it also does not contain an uncountable discrete set. (The technical term is that $\mathscr{F}[\Bbb R]$ has countable spread.) It is not true in general that countable cellularity implies countable spread, but it is true for Pixley-Roy hyperspaces. (Point $11$.)
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