1.
a) Suppose T_1 is a topology on X = {a,b,c} containing {a}, {b} but not {c}. Write down all the subsets of X which you know are definitely in T_1. Be careful not to name subsets which may or may not be in T_1.
b) Suppose T_2 is a topology on Y = {a,b,c,d,e} containing {a,b}, {b,c}, {c,d} and {d,e}. Write down all the subsets of X which you know are definitely in T_2. Be careful not to name subsets which may or may not be in T_2.
c) Invent a topology T_3 on Z = {a,b,c,d} containing {a}, {b,c} and {a,d}, but not {d}

2.
What is the only topology you can have on the one point set P = {x_0}? Describe it explicitly. Suppose (X, T) is a topological space. What is the only function you can have f: X-->P? Is it always sometimes or never continuous? Justify your answer.

3.
Let X be a topological space with at least two points and the indiscreet topology. Why is this not a topology that comes from a metric on X? Hint: you might like to try a proof by contradiction.

Solution Summary

The solution provides a review on the basic topologies first, followed by a detailed solution to the problems attached.

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