#### Jomo

##### Elite Member

- Joined
- Dec 30, 2014

- Messages
- 3,326

Any definition that I did not know I looked up.

I however do not see any errors. It's not the best written proof.

Theorem 1: Suppose X is an infinite set and

T is the finite-closed topology on X. If S is a subset X, then S is open iff

S is infinite or S is empty.

Proof: (=>)Suppose S is open in the finite-closed topology. If S is empty

we are done, so assume S is not empty. Since S is open, S is not closed, which

(by DeMorgan's law) means that S /= X and S is not finite. Thus S

is infinite.

(<=) Suppose S is either infinite or empty. If S is empty, then it is

open by denition of a topology. If S is infinite, then it is not finite so

it is not closed. Therefore S is open in the finite-closed topology.