# finite complement topology

• May 3rd 2010, 04:53 PM
bkzyankeechic
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?
• May 3rd 2010, 04:56 PM
Drexel28
Quote:

Originally Posted by bkzyankeechic
What are the closed sets in a finite space with the finite complement topology?

Come on:

Spoiler:

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

Quote:

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

Come on (with equal exasperation):

Spoiler:

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