Define a new topology on the reals

[andlook]

Homework Statement

Verify that taking \mathbb{R}, the empty set and finite sets to be closed gives a topology.

Homework Equations

The Attempt at a Solution

Clearly the empty set is finite as it has 0 elemnts, and so is closed.

If X_i, for i= {1,...,n}, are finite sets then clearly the union of finitely many finite sets is again finite and so is closed.

Let |X_i| = m_i, where m_i is the number of elements in X_i, then |\bigcap X_i | is at most max{m_i} or at least 0 if the intersection is empty. Either way it is again finite and so is defined as closed.


I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how \mathbb{R} could be included...
Thanks

 

[rasmhop]

I think I have shown the empty set, arb unions, arb intersections are closed in this topology, but I can't see how \mathbb{R} could be included...

Reread the description of the closed sets. It says:
1) That all finite sets are closed.
2) That the empty set is closed (this is really redundant, but there's no harm in including it).
3) That \mathbb{R} is closed.

 

[andlook]

Of course, as defined in the statement of the question!

Thanks