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Definition: 1.13: Boundary

A boundary is that which is the extremity of something.

Modern Definition

Let \(c\) be a figure of a plane \(p\)1. A point \(A\) is called a boundary point of \(c\), if every2 circle with the centre at \(A\) and a radius \(r > 0\) has points in common with the figure \(c\) and with \(p\setminus c\) (i.e. the everything in the plane \(p\) except the figure \(c\):

boundary_1

The set of all boundary points of \(c\) is called the boundary of the figure \(c\) and denoted by \(\delta c\).

Mentioned in:

Definitions: 1 2 3
Propositions: 4 5

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References

Bibliography

  1. Forster Otto: "Analysis 2, Differentialrechnung im \(\mathbb R^n\), Gewöhnliche Differentialgleichungen", Vieweg Studium, 1984

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Footnotes

  1. e.g. a circle ↩

  2. Requiring "every" circle means that it does not matter how small we chose the radius \(r\) of that circle, as far as it still equals some small positive number. ↩