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.