Limits in the real numbers satisfy certain algebraic properties, while the usual notion of convergence space, Hausdorff space, and topological space are purely analytic.
Let be a directed type and and be nets which converges to and respectively. By the definition of a function algebra, one could define pointwise the nets , , , , for natural number . Let us denote the limit of a net as the partial function
is a Hausdorff ring if the limit of a net partial function preserves the ring operations, provided the limit exists: