Re: Continuous functions on Munkres's book

On Mon, 30 Jul 2007, bigli wrote:

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

when the following statements are equivalent????:

This is the third time I've replied to this.

1) f is continuous

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

2 implies 1. See the proof of this in my (mars) reply to your posts
in the Ask-a-Topologist forum.

As before, the converse is false. Have you not already found an (easy)
example why the converse is false? So they aren't equivalent, unless you
add some premiss to continuous to make them so. Finding counter examples
may give hint what premiss is needed. An obvious one did not occur to me.
Maybe something about requiring f(A) to have no isolates for all A that
don't have isolates. That's my guess. You're welcome, even encouraged,
to check out my guess. Exactly when, ie can a weaker premiss be used, is
harder question.

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.

My intention from the LIMIT POINT :

A limit point of a subset E of a topological space X is defined as a
point p such that every open neighborhood V of p is such that
V-{p} intersects E.

.