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Trivial, Discrete, Cofinite, Usual and Sogenfrey Topology

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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.

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Solution Summary

Trivial, discrete, cofinite, usual and sogenfrey topology is examined.

Solution Preview

a) True
b) False. Y is given the discrete topology, then A = {1} of Y is an open set. But f^-1(A) = {0} is not open in X because X is given the ...

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