Assignment:
Let X = Y = IR and f: X -> Y be given by
f(x) = { x^2 + 1 x >= 0
{ 0 x < 0
Consider the following statement
A is open in Y then f^-1(A) is open in X.
With the following settings on X,Y, determine whether the above statement is true
a) X,Y are given the usual topology
b) X is given the trivial topology and Y is given the discrete topology
c) X is given the discrete topology and Y is given the cofinite topology
d) X is given the Sogenfrey topology and Y is given the usual topology
e) X is given the usual topology and Y is given the Sogenfrey topology.
Provide complete and step by step solution for the question and show calculations and use formulas.