Topology Spaces
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.
