Can partial indexings be totalized?

In examples like the total recursive functions or the computable real numbe rs the canonical indexings arc only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological To-spaces and study the problem in which cases the canoni cal numberings of such spaces can be totalized, i.e., have an equivalent to tal indexing. Moreover. we show under very natural assumptions that such sp aces can effectively and effectively homeomorphically be embedded into a to tally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets.