How to survive without complements

Peter Johnstone

The category of locales is often suggested as a constructive alternative to the category of topological spaces, more appropriate to the needs of domain theory (and the like). Among the comforting features one has to sacrifice in passing from spaces to locales is the fact that every subspace has a complement; however, there is a sense in which the category of locales has "enough complemented subspaces" for this not to matter. In my talk, I shall indicate how this works in practice, taking as an example a characterization of the class of open maps which is well-known for topological spaces, but which was proved for locales only this year.