What is an example of a separable Hausdorff space with a non-separable subspace?

[tylerc1991]

Homework Statement

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

The Attempt at a Solution

well since a separable space is one that is either finite or has a one-to-one correspondence with the natural numbers, the separable Hausdorff space has to be some infinite set that has a one-to-one correspondence with the natural numbers (since if I use some finite set for the separable Hausdorff set (X,T), any subset of (X,T) is also finite and therefore separable)

so the next step was trying to think of spaces that are not separable, then working backwards to think of a superset that is separable. but this is where I am getting stuck. for example, the real numbers are not separable, but any superset I can think of is also not separable. can someone give me a little push in the right direction? thank you very very much!

 

[micromass]

OK, let's first construct our space A. A must be nonseparable. Now, what nonseparable spaces do you know?

Hint: a good book in topology is the book "counterexamples in topology" by Steen and Seebach. It contains a whole lot of counterexamples to various thingies...

 

[micromass]

the real numbers are not separable

This is NOT true! The real numbers ARE separable. A countable dense subset is given by \mathbb{Q}!

 

[tylerc1991]

oh crap, I was thinking of COUNTABLE spaces. ok so that makes the problem make a lot more sense. thanks for the help!