  The efficiency of contemporary algebraic topology is not optimal since the
category of topological spaces can be made more algebraic by introducing a
profoundly new (-1)-dimensional topological space as a topological join unit.
Thereby synchronizing the category of topological spaces with the structures
within the contemporary category of simplicial complexes as well as with the
structures within the algebraic categories.
  In the category of topological spaces, the empty space {\O} has since long
been given the role as a join unit - ad-hoc though. Since it is {{\O}}, not
{\O}, that is the join unit within the category of simplicial complexes, the
role of {\O} within general topology has to be rectified.
  This article presents an algebraization of Hausdorff's century old definition
of the category of topological spaces as well as some useful algebraic
topological consequences thereof.
