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Math Help - finite compliment and discrete topology question

  1. #1
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    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?

    Please help me out
    Last edited by mr fantastic; May 22nd 2009 at 05:13 AM. Reason: Restored original question deleted by OP
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  2. #2
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    Quote Originally Posted by r2dee6 View Post
    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?
    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.
    (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)
    (c) Is f a homeomorphism? Why or why not?
    f is not a homeomorphism, because f is not continuous.
    Last edited by aliceinwonderland; April 17th 2009 at 06:27 PM. Reason: is not open->is not "necessarily" open
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