Higher separation axioms, paracompactness and

dimension are defined for semiuniform convergence spaces. Under the

assumption that subspaces are formed in the construct {\bf SUConv} of

semiuniform convergence spaces one obtains the following results:

$1^{0}$ Subspaces of normal (symmetric) topological spaces are

normal, $2^{0}$ subspaces of paracompact topological spaces are

paracompact, and $3^{0}$ for some dimension functions, the dimension

of a subspace is less than or equal to the dimension of the original

space, where these dimension functions coincide with the (Lebesgue)

covering dimension for paracompact topological spaces. Additionally,

Urysohn's Lemma, Tietze's extension theorem and some other extension

theorems are studied not only in the realm of topological spaces.

Last but not least the interrelations between nearness spaces and

semiuniform convergence spaces become apparent.