Point in Discrete Space is Neighborhood
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Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.
Let $x \in S$.
Then $\set x$ is a neighborhood of $x$ in $T$.
Proof
By definition, a neighborhood $N_x$ of $x$ is any subset of $S$ containing an open set which itself contains $x$.
That is:
- $\exists U \in \tau: x \in U \subseteq N_x \subseteq S$
From Set in Discrete Topology is Clopen we have that $\set x$ is open set in $S$.
So by Set is Subset of Itself, $\set x$ is a subset of $S$ containing an open set $\set x$ which itself contains $x$.
Hence the result.
$\blacksquare$
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: $7$