Finite Compliment Topology

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March 9th 2011, 10:57 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
March 9th 2011, 11:15 AM
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.
March 9th 2011, 11:46 AM
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