Discrete Space is Scattered
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 $H \ne \O$ is dense-in-itself.
So, again, by definition, $T$ is scattered.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness: Disconnectedness