# Homework Help: Define a new topology on the reals

1. Apr 9, 2010

### andlook

1. The problem statement, all variables and given/known data
Verify that taking $$\mathbb{R}$$, the empty set and finite sets to be closed gives a topology.

2. Relevant equations

3. 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

2. Apr 9, 2010

### rasmhop

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.

3. Apr 9, 2010

### andlook

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

Thanks