Announcing versions of papers I have been working on
recently. The first is a somewhat corrected and slightly
revised version of a fairly mature paper. (Its proofs,
equations and sections are numbered as in last year's
The second is quite new, with obviously many things
still to be learned.
The general subject of both papers is that of compatibility
between a topological (in this case metrizable) space
A and a set \Sigma of (universally quantified) equations.
(Here compatible means that \Sigma can be modeled by
continuous operations on A.) In the first case we consider
continuous operations and measure how far they must
deviate from satisfying \Sigma. In the second, we consider
exact satisfaction, but measure how far the operations
must then deviate from continuity. In either case, we
obtain a numerical measure of how badly A and \Sigma
fail to be compatible.