Fort Space is Scattered/Proof 2
Then $T$ is a scattered space.
Suppose that $H \subseteq T$ has no isolated points of $H$.
So, by definition, $H$ is dense in itself.
We have that:
So $H$ is infinite, and so contains more than one point.
So $\exists q \in H: q \ne p$.
So $H \cap \set q = \set q$ and so by definition $q$ is isolated in $H$.
Hence the result, by definition of scattered space.