A semigroup in an NR((ANR)) semigroup if it is isomorphic to the multiplicative semigroup of some near-ring (whose additive structure is commutative). A topological space X is admissible if it is Hausdorff, first countable, and for each pair of convergent sequences \(\{x_ n\}\) and \(\{y_ n\}\) where the \(\{x_ n\}\) are all distinct and all different from lim \(x_ n\), then exists a positive integer N and a continuous selfmap f of X with \(f(x_ n)=y_ n\), all \(n>N\). Admissible spaces are shown to include 0-dimensional metric spaces, and also first countable normal spaces in which each point has a neighborhood which is an absolute retract. Some sample results are (1) for admissible spaces, S(X) is an NR semigroup iff X supports a topological group structure. (2) if X is locally compact, 0-dimensional and metric, then S(X) is an NR semigroup iff X is either discrete, a Cantor set, or a product of a discrete space and a Cantor set. A result analogous to (2) is established for \(S_ p(X)\), the semigroup of continuous selfmaps of X which fix a point \(p\in X\).