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    Topology Help

    Find 3 topologies on thefive point set X = {a,b,c,d,e} such that the first is finer then the second and the second is finer then the third, without using either the trivial or discrete topology.
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    Hello,
    Quote Originally Posted by r2dee6 View Post
    Find 3 topologies on thefive point set X = {a,b,c,d,e} such that the first is finer then the second and the second is finer then the third, without using either the trivial or discrete topology.
    A topology A is said to be finer than a topology B, if any open set of B is an open set of A.

    Knowing that, it's quite easy
    Consider the less fine topology, let's say {0,{a},X} (0 designs the empty set)

    Then you have to find a topology that contains {a} (it'll always contain 0 and X, because it's the definition) and more sets.
    So you can add {b} and {a,b}. You have to add {a,b} because the union of two elements of a topology is in the topology and {a} U {b} = {a,b}
    Hence {0,{a},{b},{a,b},X} can be the second finest topology.

    You need to find a third one, that contains {a},{b},{a,b}
    Let's add {c}
    Since it has to be stable by union (for intersection, it's trivial because it's the empty set), you have to add {a} U {c}={a,c}, {b} U {c}={b,c} and {a,b} U {c}={a,b,c}
    So the finest topology can be {0,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c},X}
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