Discrete Space is not Dense-In-Itself
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Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is not dense-in-itself.
Proof
By definition, $T$ is dense-in-itself if and only if it contains no isolated points.
The result follows from Topological Space is Discrete iff All Points are Isolated.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $9$