does every infinite hausdorff space contains a countable infinite discrete subspace?

In a more general light: folklore theorem: Every infinite topological space contains a homeomorphic copy of one (or more) of the following 5 spaces:
As each of the spaces has the property that every infinite subspace of it is homeomorphic to the whole space, this list is minimal. And spaces 14 are not Hausdorff, which implies what you need, as being Hausdorff is hereditary. The nicest proof of this I know uses Ramsey's theorem (off hand I do not know a reference, who does?) using a partition of the triples or pairs of X, IIRC. Reference to the original paper: Minimal Infinite Topological Spaces, John Ginsburg and Bill Sands, The American Mathematical Monthly Vol. 86, No. 7 (Aug.  Sep., 1979), pp. 574576. The Ramsey proof I saw somewhere else, though. 


Not sure whether this is "researchlevel", but: first show that any infinite Hausdorff space has a proper infinite closed subset. Then proceed by induction. 


Yes. Lemma 1 in http://www.emis.de/journals/HOA/IJMMS/6/197.pdf 

