# Discrete Space is Scattered

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## 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

- 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