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Can an element be both boundary as well as limit point in a

Posted: Mon Feb 11, 2008 5:32 am
by mobius70
Is it possible for an element in a topology to be both boundary as well as limit point for a set?

e.g. S = {1,2,3}

Let us say 0 is null set then
T = {0, {1,2},{3},S} defines a toplogy on it.

Let is say A={1,3}

Can I have a case where a particular element of A is both limit point as well as boundary point of A.

TAKE THE CASE GIVEN ABOVE AS AN EXAMPLE ONLY.

Posted: Fri Feb 15, 2008 12:42 pm
by lime
Of course it's possible!
But before let's just in case go back and remember what actually boundary and limit point are.

The boundary of A, bd(A) is the set of all s in S such that every open set containing s intersects both A and complement of A.
A point s in S is called limit point of A if every open set that contains s also contains at least one point of A other than s.

So let's consider simple example with the set of real numbers with its standard topology. Let A be the interval (0,1). In this case we have
bd(A) = {0,1}
A' = [0,1].
Here A' - derived set- the set of all the limit points of A.
Therefore two elements of A, numbers 0 and 1 are both boundary and limit points.

In your case where
S = {1,2,3}
T = {0, {1,2},{3},S}
A={1,3}
I see that
bd(A)=0
A'=0,
so it is not really felicitous example. :wink:

Posted: Mon Feb 18, 2008 7:04 am
by mobius70
as always .. thanks for your reply .. lime