  This paper presents the Sierpinski Gasket ($\mathbb{S}$) as a final coalgebra
obtained by Cauchy completing the initial algebra for an endofunctor on the
category of tri-pointed one bounded metric spaces with continuous maps. It has
been previously observed that $\mathbb{S}$ is bi-Lipschitz equivalent to the
coalgebra obtained by completing the initial algebra, where the latter was
observed to be final when morphisms are restricted to short maps. This raised
the question "Is $\mathbb{S}$ the final coalgebra in the Lipschitz setting?".
The results of this paper show that the natural setup is to consider all
continuous functions. The description of the final coalgebra as the Cauchy
completion of the initial algebra has been explicitly used to determine the
mediating morphism from a given coalgebra to the the final coalgebra. This has
been used to show that if the structure map of a coalgebra is continuous, then
so is the mediating morphism. The description of $\mathbb{S}$ given here not
only generalizes previous observations, but also unifies classical descriptions
of $\mathbb{S}$. We also show, by means of an example, that $\mathbb{S}$ is not
the final coalgebra if we consider only Lipschitz maps.
