Question

Let X denote the real line R endowed with the trivial topology. Consider the subspace Y=Z consisting of all the integers in X. Is Y compact?

1 - Yes
2 - No
Let X denote the integers Z endowed with the cofinite topology (in YES / ONO / O 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
1 - Yes
2 - No

Answer 1

In mathematics, the integers with the trivial topology on the real line are not compact, whereas when endowed with the cofinite topology, the even integers form a compact subspace.

Step-by-step explanation:

The question relates to the topic of compactness in topology - a branch of mathematics. We are dealing with two separate cases:

1. With the trivial topology on the real line R, the subspace Y = Z (the set of all integers) is not compact. In the trivial topology, the only open sets are the empty set and the whole set, implying that any cover of Z by open sets of R cannot have a finite subcover.
2. When the set of integers Z has the cofinite topology, and we consider the subspace Y = 2Z (the set of all even integers), this subspace is compact. In the cofinite topology, every open set's complement is finite, making it straightforward to extract a finite subcover from any open cover of Y.

Therefore, in the first case, Y is not compact, but in the second case, it is compact.