Question

Let X denote the integers Z endowed with the cofinite topology (in which a subset U is deemed to be open if Z\U is finite or U=∅ ). Consider the subspace Y=2Z consisting of all even integers in Z. Is Y compact?

Answer 1

No, Y is not compact in the cofinite topology on Z because it does not have a finite open cover.

Step-by-step explanation:

In the Cofinite topology on the set of integers Z, the open sets are either finite sets or the empty set. As the subset Y=2Z consists of all even integers, it is an infinite set. Therefore, Y does not have a finite open cover. Since compactness in a topological space requires that every open cover has a finite subcover, we can conclude that Y is not compact in the cofinite topology on Z.