Re: A is dense in S

On Aug 20, 5:22 pm, Kiuhnm <kiuhnm03.4t.yahoo.it> wrote:
There is something I don't quite understand about "denseness".
For instance, I have two very similar definition:
1) A subset A of S is called dense in S if cl(A) = S, and is called
nowhere dense if S\cl(A) is dense in S.
2) A subset A of S is called dense in S if cl(A) > S. Moreover, A is
somewhere dense in S iff there exists an open U < S such that cl(A/\U) > U.

I'm also trying to prove that A is somewhere dense in S (in the sense of
2) iff int(cl(A))<>0. It's clear that if A is s.d. then int(cl(A))<>0,
but what about the converse?

Kiuhnm
****************************************************
You may be interested in the next equivalent definition of dense set
which, imo, is easier to grasp and way easier to work with in many
cases:
A subset A of a top. space X is dense (in X) iff for every open set U
in X, the intersection of A with U is non empty <==> the above is true
when U belongs to some basis of the topology on X.

Regards
Tonio

.

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