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 $$c$$:

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

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.