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.