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Math Help - Open Sets with respect to Different Topologies

  1. #1
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    Open Sets with respect to Different Topologies

    Hi, just trying to wrap my head around a few concepts; I've only just started topology so correct me if I'm wrong: is it possible, given a set X with two different topologies, to have a subset of X open with regard to one topology, and not open with regard to the other? And could someone give me an example of this please?

    Thanks!
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  2. #2
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    Re: Open Sets with respect to Different Topologies

    sure.

    here is an extreme example:

    let τ = 2X, the power set of X. in this topology EVERY subset of X is open (this is called the discrete topology on X).

    let σ = {,X}. here there are only TWO open sets, the empty set, and X itself (this is called the indiscrete topology on X).

    if X contains more than one element, then the set {x} for a single element of X, is open in the first, but not in the second.
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    Re: Open Sets with respect to Different Topologies

    Quote Originally Posted by Conn View Post
    is it possible, given a set X with two different topologies, to have a subset of X open with regard to one topology, and not open with regard to the other? And could someone give me an example of this please?
    Think about. What does it mean for the two typologies to be different?
    Keep in mind, the sets of the topology are the open sets.
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