The author considers the topological entropy of (topological) factors of a given dynamical system \((X, T)\) with positive entropy \(h\). He shows that if \(X\) is a finite-dimensional space (i.e. a space of finite topological dimension), then there exist factors with all values of entropy between 0 and \(h\). Further, in this situation, he is able to show that given one factor \((Y', S)\) of \((X, T)\), for each value of the entropy between \(h_{\text{top}} (S)\) and \(h\), there is a factor \((Y, S)\) of \((X, T)\) such that \((Y', S')\) is a factor of \((Y, S)\) and the factor maps satisfy the equation \(\varphi_{Y', Y} \circ \varphi_{Y, X}= \varphi_{Y', X}\). He shows that these results fail in infinite-dimensional systems by exhibiting an infinite-dimensional system which is minimal and of infinite topological entropy (and so has no proper factors and in particular, none of entropy between 0 and \(\infty\)).