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Math Help - topology exam

  1. #1
    Member
    Joined
    Apr 2008
    From
    Seoul, South Korea
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    128

    topology exam

    i am reviewing for topology exam coming up and i was wondering if somebody can show me how to do some review questions

    prove following statements about continuous functions and discrete and indiscrete topologies

    1. if X is discrete, then every function f from X to a topological space Y is continuous.
    2. if X is not discrete, then there is a topological space Y and a function f:X->Y that is not continuous. (hint: let Y be the set X with discrete topology)
    3. if Y is a n indiscrete topological space, then every function f from a topological space X to Y is continuous.
    4. if Y is not indiscrete, then there is a topological space X and a function f:X->Y that is not continuous.

    thank you for any help!
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  2. #2
    Member
    Joined
    May 2008
    Posts
    75
    Hi!

    1) This is trivial and follows directly from the definition of a continuous function: f continuous <=> the preimage of any open set is open. As any set is open in a discrete topology, the claim follows.

    2) Think about the hint a bit more!

    3) Follows as directly from the definitions as 1)

    4) Again it is just reading the definition.

    Banach
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