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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  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"


  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.