The topology of \(\mathbb{R}^n\)

\[\def\ld{£} \def\rn{\mathbb{R}^n} \def\half{\frac{1}{2}} \def\dd{\ \!\mathrm{d}} \def\dvvp#1#2{\frac{\partial #1}{\partial #2}} \def\dvvpn#1#2#3{\frac{\partial^#3 #1}{\partial #2^#3}} \def\of#1{\tilde{#1}} \def\ten#1{\mathsf{#1}} \def\diag#1{\mathrm{diag}{#1}} \def\abs#1{\left| #1 \right|} \def\pdv#1{\frac{\partial}{\partial #1}} \def\dv#1{\frac{\dd}{\dd #1}} \def\ddv#1#2{\frac{\dd #1}{\dd #2}} \def\vdot{\mathbf{\cdot}}\]

The space \(\rn\) is the usual \(n\)-dimensional space of vector algebra, where a position can be described by a real \(n\)-tuple, \((x_1, x_2, \dots, x_n)\).

The local topology of \(\rn\) is centred around the concept of a neighbourhood of a point in \(\rn\), and this allows the definition of a distance function between points, \(\vec{x}\) and \(\vec{y}\) in \(\rn\),

\[\label{eq:1} d(\vec{x},\vec{y}) = \left[ (x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2 \right]$^{\frac{1}{2}}$\]
neighbourhood

A neighbourhood of a point \(x\) in \(X\) is a set \(N(x)\) containing an open set which contains \(x\).

A family of neighbourhoods induces a notion of proximity to \(x\).

A neighbourhood of radius \(r\) of the point \(\vec{x}\) in \(\rn\) is thus the set of points \(N_r(\vec{x})\) with a distance less than \(r\) from \(\vec{x}\). Such a neighbourhood is discrete if each point has a neighbourhood containing no other points; which is clearly untrue for \(\rn\), which is thus continuous.

A set of points \(S\) in \(\rn\) is open if every point \(x \in S\) has a neighbourhood contained within \(S\). The interval \(a<x<b\) is an open neighbourhood, but \(a \leq x < b\) is closed, as the points \(a=x\) have neighbourhoods partially outside the set.

Hausdorff space

A space is Hausdorff if any two distinct points in the space posess disjoint neighbourhoods.

We can draw a line between two points in \(\rn\), as we can have neighbourhoods about each point which do not intersect, which is the Hausdorff property of the space.

Using the distance function \(d(\vec{x}, \vec{y})\) to define these properties induces a topology on a space, enabling the definition of open sets in the space, which have the properties

  1. if \(O_1\) and \(O_2\) are open sets, then so is their intersection, \(O_1 \cap O_2\)

  2. the union of any collection of open sets is open.

To make (1) apply to all sets the empty set must be defined as open, and for (2) to work \(\rn\) as a whole must be open.

Mappings

A map from a space \(M\) to a space \(N\) is a rule which associates elements \(x \in M\) to elements \(y \in N\). The simplest map is a function \(f:\mathbb{R} \to \mathbb{R}\) taking \(x \mapsto f(x)\).

Some terminology for mappings of the form \(f:M \to N\):

image

The set, \(T\) of elements from the set \(N\) which are mapped to from elements of \(S \subset M\). Denoted \(f(S)\).

inverse-image

The set \(S\).

There are a number of types of mapping.

many-to-one

A map where multiple elements of \(S\) map to the same element of \(T\).

one-to-one

A map where only one element of \(S\) maps to each element of \(T\).

inverse

A map which takes \(T \to S\), and is only well-defined for one-to-one maps.

Let \(f:M \to N\) and \(g: N \to P\), then the operation of function composition is defined as

\[g \circ f : M \to P\]

A map \(f:M \to N\) is continuous at \(x \in M\) if any open set of \(N\) containing \(f(x)\) contains the image of an open set of \(M\), provided \(M\),\(N\) are topological spaces. \(f\) is continuous on \(M\) if it is continuous at all the points in \(M\).

If \(f(x_1, \dots, x_n)\) is a function which is defined on an open region \(S\) of \(\rn\) the it is differentiable of class \(C^k\) if all its partial derivatives of the order less or equal to \(k\) exist and are themselves continuous on \(S\); \(f\) is a \(C^k\) function.

Real Analysis

A real function is analytic if, at \(x=x_0\) it has a Taylor expansion about \(x_0\) which converges to \(f(x)\) in a neighbourhood of \(x_0\),

(1)\[ f(x) = f(x_0) + (x-x_0) \left. \dvvp{f}{x} \right|_{x_0} + \half (x-x_0)^2 \left. \dvvpn{f}{x}{2} \right|_{x_0} + \cdots\]

This clearly requires that \(f\) be infinitely differentiable, but this is not a guarantee that \(f\) is analytic, e.g. \(\exp(-1/x^2)\) which is non-analytic as a real function, but becomes analytic with a singularity at \(z=0\) as a complex function.

A real-valued function \(g(\cdots)\) defined on an open region \(S\) of \(\rn\) is square-integrable if

\[\label{eq:3} \int_S \left[ g(x_1, \dots, x_n) \right]^2 \dd{x_1} \cdots \dd{x_n}\]

exists. A square-integrable function can be approximated by an analytic function \(g'\) such that the integral of \((g - g')^2\) over \(S\) can be made arbitrarily small.

A \(C^{\infty}\) function need not be analytic, so an analytic function is denoted \(C^{\omega}\).

An operator \(A\) acts on functions defined on \(\rn\), and takes one function, \(f\), into another one, \(A(f)\). Examples are multiplication, differentiation, and fixed-kernel integration. The commutator of two operators is defined

\[\label{eq:4} [A,B](f) = (AB-BA)(f)\]