Difference between revisions of "The Topology of Higher Dimensions: interior, closure and boundary"
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[[File:Exterior points.png|center|Exterior points]] | [[File:Exterior points.png|center|Exterior points]] | ||
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+ | ==Licensing== | ||
+ | Content obtained and/or adapted from: | ||
+ | * [http://mathonline.wikidot.com/interior-boundary-and-exterior-points-in-euclidean-space Interior, Boundary, and Exterior Points in Euclidean Space, mathonline.wikidot.com] under a CC BY-SA license |
Latest revision as of 14:33, 9 November 2021
We are now going to discuss some important classifications of points regarding a subset of which we define below.
Definition: Let . A point is said to be an Interior Point of if there exists an such that , i.e., there exists an open ball centered at for some positive radius that is a subset of . The set of all interior points of is denoted by .
In the case where and we have some subset (like the one illustrated below), then we say that a point is an interior point if there exists an open disk of some positive radius that is entirely contained in .
In the illustration above, we see that the point on the boundary of this subset is not an interior point. If we take a disk centered at this point of ANY positive radius then there will exist points in this disk that are always not contained within the pink region.
Definition: Let . A point is said to be a Boundary Point of if for every for every with there exists such that and , i.e., in every ball centered at there exists a point contained in and a point contained in the complement . The set of all boundary points of is denoted .
For , comprises the endpoints of . For , comprises the border of as illustrated below:
For , comprises the surface of .
Definition: Let . A point is said to be an Exterior Point of if . The set of all exterior points of is denoted .
For , a visualization of some exterior points of a set of points (in green) is illustrated below:
Licensing
Content obtained and/or adapted from:
- Interior, Boundary, and Exterior Points in Euclidean Space, mathonline.wikidot.com under a CC BY-SA license