Understanding Topology A on X: Real Numbers

[pivoxa15]

Homework Statement

X is the space of all real numbers

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

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.

 

[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.

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

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.

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.

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.

 

[pivoxa15]

There we go. Now you're talking.

Finally a (the first) complement from Matt Grime.

I figured that one up while typing the question.