# Finite Compliment Topology

• Mar 9th 2011, 11:57 AM
topsquark
Finite Compliment Topology
This should probably be an easy one.

The "finite compliment topology" is defined by my text as the collection of sets U of a set X such that X - U is either finite or all of X.

Nothing in the definition says that X must be finite, but I was looking at an example of a finite set and it struck me that the FCT on a finite set X is the same as the discrete topology on that same X. (The discrete topology is defined as the collection of all subsets of X.) I can verify this for a given set X but I am at a loss of how to prove it in general. I've been toying with some kind of construction or an induction proof, but can't quite seem to get a good start on it. Any thoughts?

Thanks
-Dan
• Mar 9th 2011, 12:15 PM
Drexel28
Quote:

Originally Posted by topsquark
This should probably be an easy one.

The "finite compliment topology" is defined by my text as the collection of sets U of a set X such that X - U is either finite or all of X.

Nothing in the definition says that X must be finite, but I was looking at an example of a finite set and it struck me that the FCT on a finite set X is the same as the discrete topology on that same X. (The discrete topology is defined as the collection of all subsets of X.) I can verify this for a given set X but I am at a loss of how to prove it in general. I've been toying with some kind of construction or an induction proof, but can't quite seem to get a good start on it. Any thoughts?

Thanks
-Dan

Let $U\subseteq X$ where $\#(X)<\infty$ then $\#\left(X-U\right)\leqslant \#(X)<\infty$ and so $U$ is open. Thus, the topology is just the power set, which by definition means that it's the discrete topology.

Remark: In fact, the reason that the cofinite topology is important is it is the coarsest $T_1$ topology on any set (up to homeomorphism). That said, more generally one can show that any finite $T_1$ space is discrete.
• Mar 9th 2011, 12:46 PM
topsquark
Quote:

Originally Posted by Drexel28
Let $U\subseteq X$ where $\#(X)<\infty$ then $\#\left(X-U\right)\leqslant \#(X)<\infty$ and so $U$ is open. Thus, the topology is just the power set, which by definition means that it's the discrete topology.

Remark: In fact, the reason that the cofinite topology is important is it is the coarsest $T_1$ topology on any set (up to homeomorphism). That said, more generally one can show that any finite $T_1$ space is discrete.

Very concise! Thank you.

-Dan