# finite compliment and discrete topology question

• Apr 15th 2009, 10:24 PM
r2dee6
finite compliment and discrete topology question
Let the topological space X be $\Re$ with the finite complement topology, and let the topological space Y be $\Re$ with the discrete topology. Define a function

f: X---> Y by f(x) = $\Re$ - {x}. So, for example, If
U = $\Re$ - C, where C = { ${x_1,......,x_n}$}, then f(U) = C

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

(b) What is $f^-1$?Is it continuous or not?

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

• Apr 17th 2009, 12:54 PM
aliceinwonderland
Quote:

Originally Posted by r2dee6
Let the topological space X be $\Re$ with the finite complement topology, and let the topological space Y be $\Re$ with the discrete topology. Define a function

f: X---> Y by f(x) = $\Re$ - {x}. So, for example, If
U = $\Re$ - C, where C = { ${x_1,......,x_n}$}, then f(U) = C

Quote:

(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, $f^{-1}(A)$ is not necessarily open in X.
Thus, f is not continuous.
Quote:

(b) What is $f^-1$?Is it continuous or not?
f^-1 is continuous since the domain of f^-1 is given by the discrete topology (verify this)
Quote:

(c) Is f a homeomorphism? Why or why not?
f is not a homeomorphism, because f is not continuous.