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

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mobius70
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Joined: Thu Jan 24, 2008 1:11 am

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

Post by mobius70 » Mon Feb 11, 2008 5:32 am

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.

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

Post by lime » Fri Feb 15, 2008 12:42 pm

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:

mobius70
Posts: 19
Joined: Thu Jan 24, 2008 1:11 am

Post by mobius70 » Mon Feb 18, 2008 7:04 am

as always .. thanks for your reply .. lime



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