Show that B is not a topology on R

1. Apr 11, 2013

Tomath

1. The problem statement, all variables and given/known data
Let B be the family of subsets of $\mathbb{R}$ consisting of $\mathbb{R}$ and the subsets [n,a) := {$r \in \mathbb{R} : n \leq r < a$} with $n \in \mathbb{Z}$, a $\in \mathbb{R}$ Show that B is not a topology on $\mathbb{R}$

2. Relevant equations

3. The attempt at a solution
If B were a topology then we would need:
$\emptyset$and $\mathbb{R} \in$ B (1), the arbitrary union of any opens in B to be in B (2) and any finite union of opens in B to be in B (3). Now the first two conditions (1), (2), seem to be valid so if B is not a topology on $\mathbb{R}$ then certainly condition (3) would have to fail. My question is, does condition (3) indeed fail and if it does, how can I show this?

2. Apr 11, 2013

Dick

What is the union of all of the [n,a)?

3. Apr 11, 2013

Tomath

If I am not mistaken the union of all of the [n,a) is $\mathbb{R}$

4. Apr 11, 2013

micromass

Yes. What if you take a little bit less of the $[n,a)$?? Can you form some half-open interval?

5. Apr 11, 2013

Dick

No, I don't think it's all of R. a or a+1 isn't in it. Oh, and your definition of topology is a little off. You want finite intersections to be in the topology. Specifying finite unions after you already said arbitrary union would be a little redundant.

Last edited: Apr 11, 2013