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Continuous functions on Munkres's book

  1. Jul 17, 2007 #1
    This is not a homework but it is a question in my mind.please guide me.

    Let X and Y be topological spaces,let f : X -----> Y is a function.

    when the following statements are equivalent????:

    1) f is continuous

    2) f(A') is subset of f(A)' ,for every A subset of X.

    Symbols: A' i.e limit points set of A ,and f(A)' i.e limit points set of f(A).

    pointing out: look to theorem 18-1 (page 104) from Munkres's book (TOPOLOGY 2edition 2000) and exercise 2 (page 111) from Munkres's book.
  2. jcsd
  3. Jul 17, 2007 #2


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    (2) will always imply (1). But (1) doesn't necessarily imply (2), as you probably already know. So we might try looking for extra conditions to impose on f.

    Let's suppose X and Y are arbitrary topological spaces, and f:X->Y is an arbitrary continuous function. Let A be some subset of X, and let x be in A'. We want to show that f(A') [itex]\subset[/itex] f(A)', so we want f(x) to be in f(A)'. If x sits in A, then f(x) will sit in f(A). So it would be necessary to have that f(A) [itex]\cap[/itex] f(A)' [itex]\neq \emptyset[/itex]. But this is not always true. So it appears that in order to say anything intelligent, we would have to impose some conditions on the nature of A (or X or Y), or on the topologies involved.
    Last edited: Jul 17, 2007
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