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mobius70
Joined: 24 Jan 2008
Posts: 17
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| Posted: Mon Feb 11, 2008 9:32 am Post subject: Can an element be both boundary as well as limit point in a |
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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
Joined: 04 Dec 2007
Posts: 42
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| Posted: Fri Feb 15, 2008 4:42 pm Post subject: |
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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
| Quote: | 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.  |
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mobius70
Joined: 24 Jan 2008
Posts: 17
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| Posted: Mon Feb 18, 2008 11:04 am Post subject: |
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| as always .. thanks for your reply .. lime |
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