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Math Help - Help with coarser/finer topologies

  1. #1
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    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.
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  2. #2
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    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.
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