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.
This paper gives a partial solution to a problem of W. Taylor on characterization of the unit interval in the class of all topological spaces by means of the first order properties of their clones. A characterization within the class of compact spaces is obtained.
In [5], W. Taylor shows that each particular compact polyhedron can be characterized in the class of all metrizable spaces containing an arc by means of first order properties of its clone of continuous operations. We will show that such a characterization is possible in the class of compact spaces and in the class of Hausdorff spaces containing an arc. Moreover, our characterization uses only the first order properties of the monoid of self-maps. Also, the possibility of characterizing the closed...
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