# normal topological space

• May 5th 2010, 04:24 PM
swallenberg
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.
• May 5th 2010, 04:33 PM
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
• May 5th 2010, 07:07 PM
Drexel28
Quote:

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.

Quote:

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.

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.