Open sets and limit points

  • A set A is open iff for every point a\in A, there exists a \delta such that the neighborhood \left(a-\delta,a+\delta\right) surrounding a is completely contained within (is a subset of) A.

  • A point x is a limit point of a set A iff given any \varepsilon, all neighborhoods \left(x-\varepsilon,x+\varepsilon\right) intersect the set A in some point other than x. Note that x need not be a point in A.

Let a be a point in A. Since A is open, there is a number \delta > 0 such that (a - \delta, a + \delta) \subset A.

We now show a is a limit point of A. To that end, let \epsilon > 0 be given. We may assume, without loss of generality, that \epsilon < \delta. Now, (a - \epsilon, a + \epsilon) \subset (a - \delta, a + \delta) \subset A, so (a - \epsilon, a + \epsilon) \cap A = (a - \epsilon, a + \epsilon). Since (a - \epsilon, a + \epsilon) intersects A in a point other than a, we conclude a is a limit point.

If \delta < \epsilon, then (a - \delta, a + \delta) \subset (a - \epsilon, a + \epsilon), so (a - \delta, a + \delta) \subset (a - \epsilon, a + \epsilon) \cap A). Openness already gives you an open \delta-interval around a, so you only need to worry about the case where \epsilon is smaller than that.

Cloud Advisory Services | Security Advisory Services | Data Science Advisory and Research

Specializing in high volume web and cloud application architecture, Anuj Varma’s customer base includes Fortune 100 companies (dell.com, British Petroleum, Schlumberger).

All content on this site is original and owned by AdverSite Web Holdings, Inc. – the parent company of anujvarma.com. No part of it may be reproduced without EXPLICIT consent from the owner of the content.

Anuj Varma – who has written posts on Anuj Varma, Technology Architect.


Leave a Reply

Your email address will not be published. Required fields are marked *