A topological space is discrete iff every point is its own connected component

A topological space $𝑋$ has the discrete topology iff each point $𝑥 \in 𝑋$ is in its own connected component ${𝑥}$ . #m/thm/topology

Proof

$𝑋$ is discrete iff the following function is continuous for all $𝑥 0 \in 𝑋$

𝑓 : 𝑋 \to {0, 1} 𝑥 \mapsto {0 𝑥 = 𝑥 0 1 𝑥 \neq 𝑥 0

which holds iff every $𝑥 0 \in 𝑋$ is disconnected from all other points.

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