(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Verify that taking [tex]\mathbb{R}[/tex], 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 [tex] X_i [/tex], for i= {1,...,n}, are finite sets then clearly the union of finitely many finite sets is again finite and so is closed.

Let [tex]|X_i| = m_i [/tex], where [tex]m_i[/tex] is the number of elements in [tex]X_i[/tex], then [tex]|\bigcap X_i |[/tex] is at most max{[tex]m_i[/tex]} 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 [tex] \mathbb{R}[/tex] could be included...

Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Define a new topology on the reals

**Physics Forums | Science Articles, Homework Help, Discussion**