View source for The Topology of Higher Dimensions: interior, closure and boundary

We are now going to discuss some important classifications of points regarding a subset <span class="math-inline"><math>S</math></span> of <span class="math-inline"><math>\mathbb{R}^n</math></span> which we define below.</p>
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<td><strong>Definition:</strong> Let <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span>. A point <span class="math-inline"><math>\mathbf{a} \in \mathbb{R}^n</math></span> is said to be an <strong>Interior Point</strong> of <span class="math-inline"><math>S</math></span> if there exists an <span class="math-inline"><math>r > 0</math></span> such that <span class="math-inline"><math>B(\mathbf{a}, r) \subseteq S</math></span>, i.e., there exists an open ball centered at <span class="math-inline"><math>\mathbf{a}</math></span> for some positive radius <span class="math-inline"><math>r</math></span> that is a subset of <span class="math-inline"><math>S</math></span>. The set of all interior points of <span class="math-inline"><math>S</math></span> is denoted by <span class="math-inline"><math>\mathrm{int} (S)</math></span>.</td>
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<p>In the case where <span class="math-inline"><math>n = 2</math></span> and we have some subset <span class="math-inline"><math>S \subseteq \mathbb{R}^2</math></span> (like the one illustrated below), then we say that a point <span class="math-inline"><math>\mathbf{a} \in \mathbb{R}^2</math></span> is an interior point if there exists an open disk of some positive radius <span class="math-inline"><math>r > 0</math></span> that is entirely contained in <span class="math-inline"><math>S</math></span>.</p>

[[File:Interior point.png|center|Interior point]]

<p>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.</p>
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<td><strong>Definition:</strong> Let <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span>. A point <span class="math-inline"><math>\mathbf{a} \in \mathbb{R}^n</math></span> is said to be a <strong>Boundary Point</strong> of <span class="math-inline"><math>S</math></span> if for every for every <span class="math-inline"><math>B(\mathbf{a}, r)</math></span> with <span class="math-inline"><math>r > 0</math></span> there exists <span class="math-inline"><math>\mathbf{x}, \mathbf{y} \in B(\mathbf{a}, r)</math></span> such that <span class="math-inline"><math>\mathbf{x} \in S</math></span> and <span class="math-inline"><math>\mathbf{y} \in S^c</math></span>, i.e., in every ball centered at <span class="math-inline"><math>\mathbf{a}</math></span> there exists a point contained in <span class="math-inline"><math>S</math></span> and a point contained in the complement <span class="math-inline"><math>S^c</math></span>. The set of all boundary points of <span class="math-inline"><math>S</math></span> is denoted <span class="math-inline"><math>\mathrm{bdry} (S)</math></span>.</td>
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<p>For <span class="math-inline"><math>n = 1</math></span>, <span class="math-inline"><math>\mathrm{bdry} (S)</math></span> comprises the endpoints of <span class="math-inline"><math>S</math></span>. For <span class="math-inline"><math>n = 2</math></span>, <span class="math-inline"><math>\mathrm{bdry} (S)</math></span> comprises the border of <span class="math-inline"><math>S</math></span> as illustrated below:</p>

[[File:Boundary points.png|center|Boundary points]]

<p>For <span class="math-inline"><math>n = 3</math></span>, <span class="math-inline"><math>\mathrm{bdry} (S)</math></span> comprises the surface of <span class="math-inline"><math>S</math></span>.</p>
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td><strong>Definition:</strong> Let <span class="math-inline"><math>S \subseteq \mathbb{R}^n</math></span>. A point <span class="math-inline"><math>\mathbf{a} \in \mathbb{R}^n</math></span> is said to be an <strong>Exterior Point</strong> of <span class="math-inline"><math>S</math></span> if <span class="math-inline"><math>\mathbf{a} \in S^c \setminus \mathrm{bdry} (S)</math></span>. The set of all exterior points of <span class="math-inline"><math>S</math></span> is denoted <span class="math-inline"><math>\mathrm{ext} (S)</math></span>.</td>
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<p>For <span class="math-inline"><math>n = 2</math></span>, a visualization of some exterior points of a set of points (in green) is illustrated below:</p>

[[File:Exterior points.png|center|Exterior points]]

==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