In Convenient Topology semiuniform convergences spaces including

filter spaces, uniform spaces and symmetric topological spaces as

well as their generalizations are studied in detail. Higher

separation axioms, paracompactness and dimension theory profit from

the better behaviour of subspaces of semiuniform convergence spaces

which results from their relation to nearness spaces. This has been

demonstrated by the author in an earlier paper \cite{preu3}. In the

present paper, subspaces of compact symmetric topological spaces

(resp.\ compact Hausdorff spaces) are characterized axiomatically

where the Herrlich completion of nearness spaces (resp.\ Hausdorff

completion of uniform spaces) is needed.