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Section 8.3 Topology of \(\RR^n\)

Subsection 8.3.1 What Is Topology

Topology is a branch of mathematics which studies concepts of shape, near-ness and distance. You might think that such a study woulds just be geometry, and you would be partially correct. Topology is like a coarser version of geometry. It cares about broad notions of near and far instead of precise distances, and about types of shapes instead of precisely defined specifics. Though it lacks the precision of other parts of geometry, it is fundamental to modern mathematics.

I have already shown you topology of \(\RR\) in single variable calculus, mostly notably in the definition of open and closed intervals: \((a,b)\) and \([a,b]\text{.}\) Open intervals did not include their endpoints but closed intervals did. Openness and closedness are central to topolgoy. In this brief chapter, I will extend the notions of opennes, closedness and intervals to \(\RR^n\text{.}\) I will need these notions to understand multivariable functions in the next section.

Subsection 8.3.2 Open and Closed Sets

Definition 8.3.1.

Let \(S\) be a set. A topology on \(S\) is a choice of subsets which are open and which subsets are closed.

I need two definitions which help to define open and clsoed sets in \(\RR^n\text{.}\)

Definition 8.3.2.

Let \(A\) be a subset of \(\RR^n\text{.}\) A point \(a \in A\) is called an interior point if there exists \(\epsilon > 0\) such that all points \(b\) with \(|a-b|\lt \epsilon\) are also in \(A\text{.}\)

A point is an interior point if all nearby points are also in the set. Around an interior point, small movements in any direction and remain inside the set. This \(\epsilon\) measures exactly how small the small bit of movement is.

Definition 8.3.3.

Let \(A\) be a subset of \(\RR^n\text{.}\) A point \(a\) (not necessarily in the set) is called a boundary point if for any \(\epsilon > 0\) there exists points \(b_1\) and \(b_2\) such that both \(|b_i - a|\lt \epsilon\text{,}\) \(b_1\) is in the set, but \(b_2\) isn't. The boundary of a set is the set of all its boundary point.

A point is a boundary point if there are nearby points in the set and nearby points not in the set. Nearby means within some small distance, measured by the small positive number \(\epsilon\text{.}\) The two definitions are mutually exclusive: a point cannot be both an interior and a boundary point for a set. (It can, of course, be neither). These two definitions allow me to define open and closed sets in \(\RR^n\text{.}\)

Definition 8.3.4.

Let \(A\) be a subset of \(\RR^n\text{.}\) Then \(A\) is called open is all of its points are interior points. Equivalently, \(A\) does not contin any boundary points. Alternatively, \(A\) is called closed if it contains all of its boundary points (or, equivalently, if it contains its boundary).

This definition properly extends the notions of open and closed intervals in \(\RR\text{.}\) The boundary points of an interval are the endpoints. The open intervals lacks their endpoints and the closed intervals included their endpoints. In \(\RR^n\text{,}\) the boundaries are much more complicated than endpoints: they can be intricate geometric shapes.

The sphere \(S^{n-1}\) in \(\RR^n\) is all vectors of length one. The closed ball \(B^{n}\) in \(\RR^n\) is all vectors of length one or less. The sphere \(S^{n-1}\)is the boundary of the closed ball \(B^n\text{,}\) and the closed ball is closed because all points of the sphere are all contained in the closed ball.

Subsection 8.3.3 Important Open and Closed Sets

Interval were important subsets of \(\RR\text{.}\) Here, I extend the notion to \(\RR^n\text{.}\)

Definition 8.3.6.

An open interval in \(\RR^n\) is a set of points \((x_1, x_2, \ldots, x_n) \in \RR^n\) such that

\begin{align*} \amp a_1 \amp \lt \amp x_1 \lt b_1 \\ \amp a_2 \amp \lt \amp x_2 \lt b_2\\ \amp \amp \ldots \\ \amp a_n \amp \lt \amp x_n \lt b_n \end{align*}

The notation for this interval is

\begin{equation*} I = (a_1,b_1)\times (a_2,b_2) \times \ldots \times (a_n,b_n)\text{.} \end{equation*}

Definition 8.3.7.

An closed interval in \(\RR^n\) is a set of points \((x_1, x_2, \ldots, x_n) \in \RR^n\) such that

\begin{align*} \amp a_1 \amp \leq \amp x_1 \leq b_1 \\ \amp a_2 \amp \leq \amp x_2 \leq b_2\\ \amp \amp \ldots \\ \amp a_n \amp \leq \amp x_n \leq b_n \end{align*}

The notation for this interval is

\begin{equation*} I = [a_1,b_1]\times [a_2,b_2] \times \ldots \times [a_n,b_n]\text{.} \end{equation*}

Intervals are rectangular objects. In \(\RR^2\text{,}\) they are filled-in rectangles, including the hollow bounding rectangle if the interval is closed and excluding it if the interval is open. In \(\RR^3\text{,}\) intervals are solid rectangular prisms whose boundaries are rectangular boxes. In higher dimensions, I think of intervals as higher-dimensional rectangular prisms.

Open and closed intervals are the basic open and closed sets in \(\RR^n\text{.}\) I will also define another important pair of useful open and closed sets.

Definition 8.3.8.

The open ball centred at \(v \in \RR^n\) with radius \(r\) is the set of points \(u \in \RR^n\) such that \(|u-v| \lt r\text{.}\) In \(\RR\text{,}\) this is an open interval (again, generalizing the interval). In \(\RR^2\text{,}\) this is a disc not including the boundary. In \(\RR^3\text{,}\) this is a solid sphere, again not including the boundary. In higher dimensions, this defines the higher analogue of a solid, round object without boundary.

Definition 8.3.9.

The closed ball centred at \(v \in \RR^n\) with radius \(r\) is the set of points \(u \in \RR^n\) such that \(|u-v| \leq r\text{.}\) In \(\RR\text{,}\) this is a closed interval (again, generalizing the interval). In \(\RR^2\text{,}\) this is a solid sphere including the boundary. In \(\RR^3\text{,}\) this is a solid sphere, again including the boundary. In higher dimensions, this defines the higher analogue of a solid, round object with boundary.