Ultraregular inductive limits.
We construct the example of the title.
We give an example in the Hilbert space of two subsets which are absorbing for the class of topologically complete spaces, but for which there exists no homeomorphism of onto itself mapping one of these subsets onto the other.
In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results...