8 Complete Spaces
8.1 Preamble: Equivalence
A relation \(\sim\) on a set \(S\) is an equivalence relation iff \(\forall a,b,c\in S\),
[E1]: \(a\sim a\) (reflexivity)
[E2]: \(a\sim b\Longleftrightarrow b\sim a\) (symmetry)
[E3]: \(a\sim b\,\,\&\,\,b\sim c\implies a\sim c\) (transitivity).
We can supplement this with another useful definition, the equivalence class \(\left[\cdot\right]\):
\[\forall a\in S, \left[a\right]\triangleq\left\{ b\in S|a\sim b\right\}.\]
An important result is that an equivalence relation on a set \(S\) provides a partition of \(S\) into disjoint subsets called equivalence classes. That is, it allows you to divide the elements of \(S\) into subsets with no overlapping elements (equivalence classes).
8.2 Cauchy Completeness
Consider a sequence of rational numbers \(x_{n}\in\mathbb{Q}\)
\[x\triangleq\left(\,x_{1},\,x_{2},\,x_{3},\,\ldots\,\right)\label{eq:Cauchy_seq}\]
and a metric \(d\left(x_{m},x_{n}\right)\). The sequence \(x\) is Cauchy iff:
\[\forall\epsilon\in\mathbb{Q}>0,\,\exists N,n,m\in\mathbb{N}|\forall n,m>N,\,d\left(x_{m},x_{n}\right)<\epsilon.\label{eq:Cauchy}\]
That is, for any arbitrarily small \(\epsilon\), there exists some term \(N\) in the sequence beyond which any pair of terms \(x_{m>N}\) and \(x_{n>N}\) is separated by less than \(\epsilon\). This specifies a precise sense in which the sequence converges.
The sum of two Cauchy sequences is defined term-by-term:
\[x+y\triangleq\left(\,x_{1}+y_{1},\,x_{2}+y_{2},\,x_{3}+y_{3},\,\ldots\,\right).\label{eq:Cauchy_sum}\]
Two Cauchy sequences are equivalent under the metric \(d\) iff
\[x\sim y\,\Longleftrightarrow\forall\epsilon\in\mathbb{Q}>0,\,\exists N,n\in\mathbb{N}|\forall n>N,\,d\left(x_{n},y_{n}\right)<\epsilon.\label{eq:cauchy equivalence}\]
This defines a precise sense in which the sequences converge to the same limit (equivalently, their difference converges to zero). Here, \(\sim\) is a formal equivalence relation obeying E1-E3. Hence, Cauchy sequences defined on a set \(S\) divide the set into disjoint equivalence classes.
8.2.1 Example: \(\mathbb{Q}\hookrightarrow\mathbb{R}\)
The set of rationals \(\mathbb{Q}\) is ‘incomplete’. For example, polynomials such as
\[x^{2}-2=0\label{eq:quadratic}\]
have rational coefficients but no rational solutions.
Infinite series/sums of rationals can be defined which tend/evaluate to things which are not rational.
However, Cauchy sequences give a formal method of completing the rationals. The completion of a space is denoted \(\hookrightarrow\). Consider the set \(X\) of all Cauchy sequences with elements in \(\mathbb{Q}\). Choosing \(d\) to be the usual Euclidean metric provides a unique partition of \(X\) into equivalence classes. We define (!) this partition to be \(\mathbb{R}\).
Hence we have the definition of real numbers (\(\mathbb{R}\)):
\(\mathbb{R}\) is the set of all equivalence classes of Cauchy sequences in \(\mathbb{Q}\) under the Euclidean metric.
For example, \(e\) is the equivalence class of the Cauchy sequence
\[e=\left(2,\,2.7,\,2.71,\,2.718,\,2.7182,\,\ldots\right).\label{eq:e}\]
An equivalent sequence, also in this equivalence class, is defined by
\[x_{n>0}=\left(1+1/n\right)^{n}:\,\,\left(2,\,9/4,\,64/27,\,\ldots\right)\label{eq:e_2}\]
and another by
\[x_{n>0}=\sum^{n}_{i=0}\frac{1}{i!}:\,\,\left(2,\,5/2,\,8/3,\,\ldots\right).\label{eq:e_3}\]
Note that any rational \(q\) can be represented as a Cauchy sequence simply by writing
\[q=\left(q,\,q,\,q,\,\ldots\right)\label{eq:Caucy_Q}\]
hence \(\mathbb{Q}\subset\mathbb{R}\).
8.3 Definition of a Complete Metric Space
A metric space \(\mathcal{M}\) is complete iff every Cauchy sequence of points in \(\mathcal{M}\) converges to a point in \(\mathcal{M}\).