Discrete Space is Scattered

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.

Then $T$ is a scattered space.


Proof

We have that Topological Space is Discrete iff All Points are Isolated.

So, by definition, no subset $H \subseteq S$ of $T$ such that $H \ne \varnothing$ is dense-in-itself.

So, again, by definition, $T$ is scattered.

$\blacksquare$


Sources