# Thread: normal topological space

1. ## normal topological space

Hello!

Ok, here it goes:

T={[x,y); x,y belong to R and x<y} (R = the set of real numbers)

Show that (R,T) is normal.

2. I don't understand the queston because T doesn't seem to be a topology may be T is the base for a topology on R

3. Originally Posted by swallenberg
Hello!

Ok, here it goes:

T={[x,y); x,y belong to R and x<y} (R = the set of real numbers)

Show that (R,T) is normal.
Originally Posted by facenian
I don't understand the queston because T doesn't seem to be a topology may be T is the base for a topology on R
It's supposed to be the topology for which that is a base, it is usually called the half-open interval topology or the Sorgenfrey line.

So, how much topology do you know? How much do you know about this specific topology?

Where's your work?

It's clearly Hausdorff, right? Since if $\displaystyle x\ne y$ then WLOG $\displaystyle x<y\implies x<\frac{x+y}{2}<y$ and so $\displaystyle [x,\tfrac{x+y}{2}),(\tfrac{x+y}{2},y]$ are disjoint basic open sets containing them.