Completeness

Definition

A Metric Space $(X,\mathcal{T},d)$ is complete If and only if every Cauchy Sequence on $X$ converges to an element of $X$.

Facts

Consider a complete Metric Space $(X,\mathcal{T},d)$ and a subspace $A\subset X$. The subspace $A$ is complete if and only if $A$ is a closed set.