Re: closed sets, limit points

On 28 May 2007 07:02:43 -0700, frank <frank.degeeter@xxxxxxxxx> wrote:

A limit point is a point every deleted neighbourhood of which contains
at least one point of C. Now, it is possible that each of these
neighbourhoods contain elements outside C as well. In that case, the
limit point is a boundary point. If not, some neigbourhoods are
completely contained in C. The limit point then is an internal point.

For future reference, please don't top post.

José Carlos Santos schreef:
On 28-05-2007 11:01, frank wrote:

A set C in R^n is closed if and only if it contains all its limit
points. My analysis book has a proof of this theorem. While pondering
over it, I possible devised another proof, but I would be thankful if
someone would have a look at it and tell me whether it is correct - I
am suspicious because it looks a bit too simple.

Here it goes:
If the set is closed, by definition it contains all its boundary
points. But any boundary point is also a limit point. Limit points
which are not boundary points are internal points, and therefore also
contained in the set. Therefore, it the set is closed, it contains all
its limit points.
If a set contains all its limit points, it also contains its boundary
points, because these are limit points as well. But then, by
definition, the set is closed.

Right?

How do you prove that "[l]imit points which are not boundary points are
internal points"? Not that I doubt that it is true.

Best regards,

Jose Carlos Santos

Your proof looks fine to me.

Note, a closed set is often defined as a set which contains all its
limit points. One can then prove (in the same way as you did) that
such a definition is equivalent to the one in your book.

quasi
.

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