- #1
tylerc1991
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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!