# Help with coarser/finer topologies

• August 6th 2008, 03:11 AM
pruzchett
Help with coarser/finer topologies
Consider the usual topology on R (set of real nos).

U = {V R: if x V then an open interval (a,b) s.t. x (a,b) V}
and consider the left-hand topology on R..

L = {
V R: if x V then a,b V, a < b, s.t. x [a,b) V}

does it follow that the left hand topology is finer than the usual topology? Is any U-open set also L-open? I'm confused.

Is (0,1), which is U-open, also L-open?

Can someone offer enlightenment? Thanks.
• August 6th 2008, 06:10 AM
Opalg
The equation $(a,b) = \bigcup_{n=1}^{\infty}[a+{\textstyle\frac1n},b)$ shows that every open interval is a union of L-open sets, and hence is open in the L topology. Since the open intervals generate the usual topology, it follows that U⊆L (in other words, the left-hand topology is finer than the usual topology).

In particular, if you put a=0 and b=1, the same equation tells you that (0,1) is L-open.