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This paper presents the Sierpinski Gasket ($\mathbb{S}$) as a final coalgebraobtained by Cauchy completing the initial algebra for an endofunctor on thecategory of tri-pointed one bounded metric spaces with continuous maps. It hasbeen previously observed that $\mathbb{S}$ is bi-Lipschitz equivalent to thecoalgebra obtained by completing the initial algebra, where the latter wasobserved to be final when morphisms are restricted to short maps. This raisedthe question "Is $\mathbb{S}$ the final coalgebra in the Lipschitz setting?".The results of this paper show that the natural setup is to consider allcontinuous functions. The description of the final coalgebra as the Cauchycompletion of the initial algebra has been explicitly used to determine themediating morphism from a given coalgebra to the the final coalgebra. This hasbeen used to show that if the structure map of a coalgebra is continuous, thenso is the mediating morphism. The description of $\mathbb{S}$ given here notonly generalizes previous observations, but also unifies classical descriptionsof $\mathbb{S}$. We also show, by means of an example, that $\mathbb{S}$ is notthe final coalgebra if we consider only Lipschitz maps.

Sierpinski Gasket as a Final Coalgebra Obtained by Cauchy Completing the Initial Algebra