A note on almost periodic points and minimal sets in - and -spaces.
The clone of a topological space is known to have a strictly more expressive first-order language than that of the monoid of continuous self-maps. The current paper studies coclones of topological spaces (i.e. clones in the category dual to that of topological spaces and continuous maps) and proves that, in contrast to clones, the first-order properties of coclones cannot express anything more than those of the monoid, except for the case of discrete and indiscrete spaces.
A condensation is a one-to-one continuous mapping onto. It is shown that the space of real-valued continuous functions on in the topology of pointwise convergence very often cannot be condensed onto a compact Hausdorff space. In particular, this is so for any non-metrizable Eberlein compactum (Theorem 19). However, there exists a non-metrizable compactum such that condenses onto a metrizable compactum (Theorem 10). Several curious open problems are formulated.