# Why not a topology?

1. Sep 19, 2007

### pivoxa15

1. The problem statement, all variables and given/known data
X is the space of all real numbers

topology A={empty set} U {R} U {(-infinity,x];x in R}

3. The attempt at a solution
Is it because (-infinity, x] is not an open set usuing the usual metric on R but is using a metric allowed as it was not specified in the question.

If not then is it because an infinite union of (-infinity,x] is not in A?

i.e take x=1/n then an infinite union of (-infinity,-1/n] when n goes to infinity should be (-infinity,0) which is not in A.

2. Sep 19, 2007

### matt grime

What does the usual topology have to do with anything. You're asked to show that this is not a topology so find one of the axioms it fails to satisfy.

that doesn't make sense. The union of what index? Do you mean the union

$$\cup_{x \in \mathbb{R}} (-\infty,x]$$?

Because that is just R.

There we go. Now you're talking, though what you really mean is

$$\cup_{n \in \mathbb{N}} (-\infty,-1/n]=(-\infty,0)$$.

There are no limits of anything involved.

Last edited: Sep 19, 2007
3. Sep 19, 2007

### pivoxa15

Finally a (the first) complement from Matt Grime.

I figured that one up while typing the question.