Clones and coclones motivate this examination of coconnected spaces. A space $X$ is coconnected if every continuous map $X\times X\to X$ depends only on one variable. We prove here that every monoid can be represented as the monoid of all nonconstant continuous selfmaps of a coconnected space and that, within the class of Hausdorff spaces, the coconnectedness is not expressible by a sentence of the first order language of the monoid theory: we construct two Hausdorff spaces with isomorphic monoids of all continuous selfmaps such that one of them is coconnected and the other is not.