Clones of continuous maps of topological spaces and clones of homomorphisms of universal algebras are investigated and their initial segments compared. We show, for instance, that for every triple $2\leq n_1\leq n_2\leq n_3$ of integers there exist algebras $\caa_1$ and $\caa_2$ with two unary operations such that the initial $k$-segments of their clones of homomorphisms are equal exactly when $k\leq n_1$, isomorphic exactly when $k\leq n_2$ and elementarily equivalent exactly when $k\leq n_3$.