Proof That an Infinite Subset of a Discrete Space is Not Compact
An infinite subsetof a discrete spaceis not compact.
Letbe a discrete topological space and letbe an infinite subset ofConsider the set of singleton sets
It is an open cover ofbecause andis open inbecause every singleton set is open in the discrete topology.
No proper subset ofis an open cover of A. Since A is infinite so isand the open cover contains no fininite subcover andis not compact.
Obviously ifis a finite subset of a discrete space it is compact.