Regular funcoids, a generalization of regular topospaces (rewritten because of an error)

Both my definition and description of properties of regular funcoids were erroneous. (The definition was not compatible with the customary definition of regular topological spaces due an error in the definition, and its properties included mathematical errors.)

I have rewritten the erroneous section of my book.

Now it is shown that being regular for a funcoid $latex f$ is equivalent to each of the following formulas:

1. $latex \mathrm{Compl}\,(f \circ f^{-1} \circ f) \sqsubseteq \mathrm{Compl}\,f$.
2. $latex \mathrm{Compl}\,(f \circ f^{-1} \circ f) \sqsubseteq f$.

These formulas seem not being an example of math beautify. So I suspect that the traditional definition of regular topospaces should be amended (or rather not to produce a terminology conflict, replaced with an other algebraically more elegant concept).