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

Forum for the GRE subject test in mathematics.
mobius70
Posts: 19
Joined: Thu Jan 24, 2008 1:11 am

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

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.

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am
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.

so it is not really felicitous example. 