Hi all again.
I have a very simple question that I cannot find a clue.
(external of ), where is the closure and is the complementary.
(boundary of ).
Prove that .
Thanks.
Hi all again.
I have a very simple question that I cannot find a clue.
(external of ), where is the closure and is the complementary.
(boundary of ).
Prove that .
Thanks.
A point if and only if each open set that contains contains a point in and a point in .
Read that definition carefully several times.
It follows from that definition that any such point is a boundary point of both and , recall that .
If is the whole space then .
Thanks Plato, but I guess I need more help.
For instance, do we have ?
I could not find a counter example.
If so, I have a solution as follows.
.............._
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.............._
.............._
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.............._
Is that true?
Note. We work on .
A slight variation on what Plato said: if you define the interior points of a set, A, as you do in metric topology- points, p, such that some neighborhood of p is a subset of A- then you can define the exterior of A as the set of all interior points of the complement of A and the boundary points of A as the set of all points in the space that are neither interior points nor exterior points of A. Because "complement of complement of A" is A, The interior points of A are the exterior points of and vice versa. What does that tell you about the boundary points of A.