Every Hausdorff topological linear space is a Hausdorff

topological group, which is a completely regular (Hausdorff) space by Theorem 5 in [7], p.

For a Tychonoff space X, we denote by L(X), V(X), F(X), and A(X) the free locally convex space, the free topological vector space, the free

topological group, and the free abelian

topological group over X, respectively.

For a topological field k, GL(k) = [[union].sub.n[right arrow]m][GL.sub.n](k) is a

topological group with the direct limit topology, that is, a subset U of GL(fc) is open if and only if U [intersection] [GL.sub.n](k) is open for each n [greater than or equal to] 1 (e.g., 3.1 of [4]).

When S is a

topological group, then LUC(S) is the set of all uniformly continuous functions on S with respect to the right uniformity of S i.e., f [member of] LUC(S) [??] [for all] [epsilon] > 0, [there exists] U neighborhood of the identity of S such that [s.sup.-1]t [member of] U [??] [absolute value of f (s) - f (t)] [less than or equal to] [epsilon].

If we set S := G x T, where G is an abelian

topological group, then S is a reflexive foundation semigroup and again by Theorem 4.1, [M.sub.a](S) is BSE.

Keywords: Semi open set, semi closed set, irresolute mapping, semi homeomorphism, irresolute

topological group, semi connected space, semi component, semi

topological groups with respect to irresoluteness.

In studying the strong topologies on the generalized PN spaces, Alsina et al., 1997 [3] investigated the continuity of the probabilistic norm; they pointed out that each PN space is a

topological group but may not be a topological vector space.

Let G be a

topological group. Given two continuous representations V and V' of G, we are interested in determining up to what extent they are "independent".

It is well known that there exists a one to one correspondence between Alexandroff topologies on group [member of] which made [member of] a

topological group and its normal subgroups ([4], theorem 4), moreover if for each normal subgroup N of G, [T.sub.N] denotes the Alexandroff topology on G obtained from N, then [T.sub.N] is the topology with minimal basis {gN : g [member of] G}.

Namely, consider the category of topological G-modules as a subcategory of the classifying topos BG of G (continuous cohomology of a

topological group G, as in S.

Remainders of a

topological group or a paratopological group have many interesting properties and have been studied extensively in literature (see [1]-[6] and [8]-[11]).

They begin with real numbers, describing real numbers as an ordered set and explaining the concept of real numbers as a field and as an ordered group and a

topological group. They explain complex numbers and rational numbers, expanding on the latter as a field and describing rational numbers as a field, then work through the concept of completion, as chains, ordered groups, topological abelian groups, and topological rings and fields.