Difference between revisions of "The Topology of Higher Dimensions: interior, closure and boundary"

Revision as of 14:27, 9 November 2021

We are now going to discuss some important classifications of points regarding a subset $S$ of $\mathbb {R} ^{n}$ which we define below.

Definition: Let $S\subseteq \mathbb {R} ^{n}$ . A point $\mathbf {a} \in \mathbb {R} ^{n}$ is said to be an Interior Point of $S$ if there exists an $r>0$ such that $B(\mathbf {a} ,r)\subseteq S$ , i.e., there exists an open ball centered at $\mathbf {a}$ for some positive radius $r$ that is a subset of $S$ . The set of all interior points of $S$ is denoted by $\mathrm {int} (S)$ .

In the case where $n=2$ and we have some subset $S\subseteq \mathbb {R} ^{2}$ (like the one illustrated below), then we say that a point $\mathbf {a} \in \mathbb {R} ^{2}$ is an interior point if there exists an open disk of some positive radius $r>0$ that is entirely contained in $S$ .

Interior point

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 $S\subseteq \mathbb {R} ^{n}$ . A point $\mathbf {a} \in \mathbb {R} ^{n}$ is said to be a Boundary Point of $S$ if for every for every $B(\mathbf {a} ,r)$ with $r>0$ there exists $\mathbf {x} ,\mathbf {y} \in B(\mathbf {a} ,r)$ such that $\mathbf {x} \in S$ and $\mathbf {y} \in S^{c}$ , i.e., in every ball centered at $\mathbf {a}$ there exists a point contained in $S$ and a point contained in the complement $S^{c}$ . The set of all boundary points of $S$ is denoted $\mathrm {bdry} (S)$ .

For $n=1$ , $\mathrm {bdry} (S)$ comprises the endpoints of $S$ . For $n=2$ , $\mathrm {bdry} (S)$ comprises the border of $S$ as illustrated below:

Boundary points

For $n=3$ , $\mathrm {bdry} (S)$ comprises the surface of $S$ .

Definition: Let $S\subseteq \mathbb {R} ^{n}$ . A point $\mathbf {a} \in \mathbb {R} ^{n}$ is said to be an Exterior Point of $S$ if $\mathbf {a} \in S^{c}\setminus \mathrm {bdry} (S)$ . The set of all exterior points of $S$ is denoted $\mathrm {ext} (S)$ .

For $n=2$ , a visualization of some exterior points of a set of points (in green) is illustrated below:

Exterior points

@@ Line 1: / Line 1: @@
 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>
 <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>Interior Point</strong> of <span class="math-inline"><math>S</math></span> if there exists an <span class="math-inline"><math>r &gt; 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>
+<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>
 </blockquote>
-<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 &gt; 0</math></span> that is entirely contained in <span class="math-inline"><math>S</math></span>.</p>
+<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>
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%205.43.29%20PM.png" alt="Screen%20Shot%202015-09-13%20at%205.43.29%20PM.png" class="image" /></div>
+[[File:Interior point.png|thumb|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>
 <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 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 &gt; 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>
+<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>
 </blockquote>
 <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>
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%206.02.25%20PM.png" alt="Screen%20Shot%202015-09-13%20at%206.02.25%20PM.png" class="image" /></div>
+[[File:Boundary points.png|thumb|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;">
@@ Line 16: / Line 20: @@
 </blockquote>
 <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>
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/interior-boundary-and-exterior-points-in-euclidean-space/Screen%20Shot%202015-09-13%20at%206.02.07%20PM.png" alt="Screen%20Shot%202015-09-13%20at%206.02.07%20PM.png" class="image" /></div>
+[[File:Exterior points.png|thumb|Exterior points]]

Difference between revisions of "The Topology of Higher Dimensions: interior, closure and boundary"

Revision as of 14:27, 9 November 2021

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Tools