# 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 \$\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.

Quote:

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