Thread: finite compliment and discrete topology question

Let the topological space X be with the finite complement topology, and let the topological space Y be with the discrete topology. Define a function

f: X---> Y by f(x) = - {x}. So, for example, If
U = - C, where C = { }, then f(U) = C

(a) Is f continuous?Why or why not?

(b) What is ?Is it continuous or not?

(c) Is f a homeomorphism? Why or why not?

Please help me out

Originally Posted by r2dee6

Let the topological space X be with the finite complement topology, and let the topological space Y be with the discrete topology. Define a function

f: X---> Y by f(x) = - {x}. So, for example, If
U = - C, where C = { }, then f(U) = C

(a) Is f continuous?Why or why not?

Let A be an infinite union of singleton sets in the topological space Y. A is open in the topological space Y.
However, is not necessarily open in X.
Thus, f is not continuous.

(b) What is ?Is it continuous or not?

f^-1 is continuous since the domain of f^-1 is given by the discrete topology (verify this)

(c) Is f a homeomorphism? Why or why not?

f is not a homeomorphism, because f is not continuous.