Completion

2020 Mathematics Subject Classification: Primary: 54E50 [MSN][ZBL]

of a metric space $(X,d)$

Given a metric space $(X,d)$, a completion of $X$ is a triple $(Y,\rho ,i)$ such that:

• $(Y,\rho )$ is a complete metric space;
• $i:X\to Y$ is an isometric embedding, namely a map such that $d(x,y)=\rho (i(x),i(y))$ for any pair of points $x,y\in X$;
• $i(X)$ is dense in $Y$.

Often people refer to the metric space $(Y,\rho )$ as the completion. Both the space and the isometric embedding are unique up to isometries.

The standard construction of the completion is through Cauchy sequences and can be described as follows. Consider the set $Z$ of all possible Cauchy sequences $\{x_k\}$ of $X$ and introduce the equivalence relation $$\{x_k\}\sim \{y_k\}⟺\underset{k\to \mathrm{\infty }}{lim}d(x_k,y_k)=0.$$ $Y$ is then the quotient space $Z/\sim$ endowed with the metric $$\rho ([\{x_k\}],[\{y_k\}])=\underset{k\to \mathrm{\infty }}{lim}d(x_k,y_k).$$ The map $i:X\to Y$ maps each element $x\in X$ in the constant sequence $x_n=x$.

A notion of completion can be introduced in general uniform spaces: the completion of a metric space is then just a special example. Another notable special example is than the completion of a topological vector space.

The concept of completion was first introduced by Cantor, who defined the space of real numbers as the completion of that of rational numbers, see real number

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