What is a separable Hausdorff subspace and how can it be demonstrated?

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Homework Statement



Give an example of a topological space (X,T) that is separable and Hausdorff, with a subspace (A,T_A) that is not separable.

The Attempt at a Solution



Let X = R and T be a topology on X whose basis elements are open intervals intersected with the rationals and individual irrational numbers, essentially the discrete topology on the irrationals. Then T is Hausdorff and separable, and has a subspace (the irrationals) that is not separable.
 
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Why do you think that space is separable? What is the countable dense subset?
 

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