Category:Completion Theorem (Metric Space)

This category contains pages concerning Completion Theorem (Metric Space):

Let $M = \struct {A, d}$ be a metric space.

Then there exists a completion $\tilde M = \struct {\tilde A, \tilde d}$ of $\struct {A, d}$.

Moreover, this completion is unique up to isometry.

That is, if $\struct {\hat A, \hat d}$ is another completion of $\struct {A, d}$, then there is a bijection $\tau: \tilde A \leftrightarrow \hat A$ such that:

$(1): \quad \tau$ restricts to the identity on $x$:

$\forall x \in A: \map \tau x = x$

$(2): \quad \tau$ preserves metrics:

$\forall x_1, x_2 \in A : \map {\hat d} {\map \tau {x_1}, \map \tau {x_2} } = \map {\tilde d} {x_1, x_2}$