This is Intro Analysis proof. Yes, K subset R. I am trying to prove it using basic Intro Analysis techniques. I presume that theorems like this one were proven before Topology became a major area of mathematics. Does it not seem like cheating (proving Analysis theorems using techniques from Topology given that these theorems were proven without using techniques from Topology)?
Not really, Hausdorff is just a property that spaces have, it's not really cheating. Someone proves something in like
then notices that they didn't really CARE that they were inside the real numbers, all they cared about was that they were able to separate two points by open sets, and defined Hausdorff spaces to be the places where you can do that.
In other news: PLEASE include ALL assumptions in the future, this could have been vastly simplified if we had known that
from the start rather than just an abstract compact set in some ambient space.
My post which starts with "Well, you'll at least need that your ambient space is Hausdorff[. . .]" has what you need, you'll need to show that you can always separate two points in
by neighborhoods, but that is easy, just choose
and then the
doesn't intersect the
, then proceed with the rest of the proof that the complement is open.