# Thread: finite complement topology

1. ## finite complement topology

What are the closed sets in a finite space with the finite complement topology?

What are the closed sets in a infinite space with the finite complement topology?

2. Originally Posted by bkzyankeechic
What are the closed sets in a finite space with the finite complement topology?
Come on:

Spoiler:

If $\displaystyle X$ is finite then given any $\displaystyle E\subseteq X$ we have that $\displaystyle X-E\subseteq X$ is finite and thus $\displaystyle E$ is open. Thus, $\displaystyle X$ is discrete...sooo.

What are the closed sets in a infinite space with the finite complement topology
Come on (with equal exasperation):

Spoiler:

If $\displaystyle X$ is infinite and $\displaystyle E\subseteq X$ is closed then $\displaystyle E=X-O$ for some open $\displaystyle O\subseteq X$ but by definition this means that $\displaystyle E$ is finite