A space X is called K-analytic if there is an upper semicontinuous compact-valued map T from the product space [N.sup.N], where N is equipped with the

discrete topology, into X such that [universa][T([alpha]) : [alpha] [member of] [N.sup.N]} = X.

The

discrete topology on a set is a topology having the desired property.

But [tau] is

discrete topology so [tau] = {[phi],X,{a},{b}} and family of all [alpha]-open sets is [T.sub.[alpha]] = {[phi],X,{a},{b}}.

Let [E.sup.[delta]] denote E endowed with the

discrete topology; the identity map on the underlying set provides a continuous [GAMMA]-equivariant map [E.sup.[delta]] [right arrow] E.

For example, there is one [T.sub.0]-topology with one level (the

discrete topology) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [T.sub.0]-topologies with two levels.

The

discrete topology of IFRSs in X contains all the IFRSs in X.

In a similar manner, each of the other elements of X can be written as the intersection of two semi-open sets and, thus, (X,T) has the pairwise semi-open intersection property and TSO(X,T) is the

discrete topology on X.

the strong topology on this PN space is the

discrete topology on the set [R.sup.n].

Now suppose that K has the

discrete topology. The subsets [F.sub.n]([V.sub.e]) define a fundamental system of neighbourhoods of zero of a Hausdorff K-vector topology on V = [V.sub.e] (see [12]).

Example 3.3 Let X = (0, 1], we define a topology [[tau].sub.1]-

discrete topology and [[tau].sub.2] = {[phi], X, (a,1], [alpha] [member of] X}.

We will think of g[.sub.Q] and [G.sub.Q] as having the

discrete topology. [G.sub.Q] has no normal abelian group of finite index; therefore [G.sub.Q] is not of Type I ([Tho]).