Exercise 2.8

Exercise 2.8: Let $$X$$ be any set, and let $$\mathcal{T}$$ consist of the empty set and all subsets of $$X$$ whose complement in $$X$$ is finite. Show that $$\mathcal{T}$$ is a topology. (This is known as the cofinite topology.)

Olomana
$$\mathcal{T}$$ satisfies the definition of a topology:

i) $$\emptyset \in \mathcal{T}$$ by definition.

ii) $$X \in \mathcal{T}$$ since the complement of $$X$$ is $$\emptyset$$, which is finite.

iii) The union of any collection of members of $$\mathcal{T}$$ is also a member of $$\mathcal{T}$$:

Proof of (iii): Let $$U$$ be the union of any number of sets $$U_i$$ in $$T$$.

By De Morgan's Law, the complement of $$U$$ is equal to the intersection of the complements of $$U_i$$. By definition, these complements are all finite, and the intersection of any number of finite sets is also finite. Since the complement of $$U$$ is finite, $$U$$ is in $$T$$.

iv) The intersection of any two members of $$T$$ is also a member of $$T$$:

Proof of (iv): Let $$U$$ be the intersection of $$U_1$$ and $$U_2$$ in $$T$$.

By De Morgan's Law, the complement of $$U$$ is equal to the union of the complements of $$U_1$$ and $$U_2$$. By definition, these complements are both finite, and the union of two finite sets is also finite. Since the complement of $$U$$ is finite, $$U$$ is in $$T$$.

Olomana 21:49, March 14, 2011 (UTC)

'''Good. I linked to where someone can make a page for De Morgan's Law since it's such a useful set theory concept that we all need to be familiar with. --Steven.clontz 19:14, March 17, 2011 (UTC)'''