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Definition A seperable space $X$ is a topological space which contains a countable subset $A$ dense in $X$. For example, $\mathbb{R}$ is separable as $\mathbb{Q}$ is a countable subset of $\mathbb{R}$ and $\bar{\mathbb{Q}}=\mathbb{R}$.

Now I need to show that a subspace of a separable space is not necessarily a separable one. I guess the subspace of all irrational numbers $I:=\mathbb{R}-\mathbb{Q}\subset \mathbb{R}$ may work it out, but I just don't know how to get a contradiction if $I$ has a countable dense subset. Also it seems vague to me what topology is appropriate for $I$, the one inherited from open intervals or 'left topology' with basis $[a,b)$, or something else...?

Any help would be appreciated! Thanks!

atlantic0cean
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1 Answers1

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The real numbers with the discrete topology are not separable because each point is open.

We can make it separable by deforming it a bit. Try to add an element $x$ and make the topology so that the open sets all have to contain $x$ (that is a set is open if and only if it contains $x$ (or is the open set)). I think this is a topological space.

This topological space is separable because $\{x\}$ is a countable dense subset. However the subspace topology on $\mathbb R$ is not,because it is the discrete topology.

Asinomás
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