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