Jason (jcreed) wrote,
Jason
jcreed

-

Progressing on topo paper ideas.
I found conditions paramatrized by cardinals
strictly between T_2 and T_3. A space is k-separatable
if any cardinality-k collection of isolated points
can be separated by disjoint open sets.
So T_1 and 2-separatable => Hausdorff.
Also if a space is countably based and aleph_0-separatable,
I think it must be regular. Perhaps more generally if
a space has a k-basis and is k-separatable, it's regular?

Hm, now that I think about it, this claim is not at all
obvious. I've come pretty close to finding an irregular
w-based w-separatable space, but not quite.

(Erg, no that counterexample doesn't actually work)
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