Continuous functions on Munkres's book

[bigli]

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.

 

[morphism]

(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') \subset 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) \cap f(A)' \neq \emptyset. 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.