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!
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.