Closed discrete subset

A closed discrete subset $D$ of a topological space $X$ is a closed set whose subspace topology is the discrete topology. Every point of $D$ therefore has a neighbourhood meeting $D$ at that point alone. Closedness excludes all accumulation points outside $D$ as well.

In a plane domain, a closed discrete subset is exactly a set meeting every compact subset in finitely many points. This local finiteness is necessary for a nonzero holomorphic function to have the subset as its entire zero set. A merely isolated-point subset can have an accumulation point outside the subset but inside the domain, as $\{1/n:n\ge 1\}\subset \mathbb{C}$ does.

// Destination: analysis.bigb

Let $D$ be a closed discrete subset of a compact space $X$. Closedness makes $D$ compact. Discreteness makes its singletons open in its subspace topology, so the singleton cover has a finite subcover. Thus $D$ is finite. Closedness is essential: the discrete set $\{1/n:n\ge 1\}$ inside the compact interval $[0,1]$ is infinite and has the missing accumulation point zero.