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The problem is sketch "boundary of any finite set of points in the plane" and determine its boundary. I'm totally confused since there'are infinitely finite sets in the plane, such as one point alone. How can I sketch its picu

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Don't you mean convex envelope of the points instead of topological boundary? – Berci Sep 30 '12 at 11:34

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There isn't one possible picture, but they are all about equally interesting. The boundary is the set of points in the plane which have both points in and out of your finite point-set 'near' them. Since a finite point set is exclusively made up of separated points (the interior of the set is empty) the boundary is empty.

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Wrong: the boundary is the set itself. – Robert Israel Sep 30 '12 at 6:46
Why isn't points themselves boundaries? – Frank Xu Sep 30 '12 at 6:53
@Frank: They are indeed part of the boundary. If $F$ is a finite set in the plane, its boundary is itself $F$. The problem is asking you simply to sketch some finite set and indicate its boundary; there are, as you say, infinitely many different pictures that you could correctly sketch. – Brian M. Scott Sep 30 '12 at 6:58
OK. Got it, thank you guys. – Frank Xu Sep 30 '12 at 7:02

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