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.

Let us recall (see [1]) that the free locally convex space L(X) on a Tychonoff space X is a pair consisting of a locally convex space L(X) and a continuous map i : X [right arrow] L(X) such that every continuous map f from X to any locally convex space E gives rise to a unique continuous linear operator [sup.f] : L(X) [right arrow] E with f = [bar.f] [omicron] i.

Recall that a Tychonoff space X is called Dieudonne complete if its topology is induced by a complete uniformity.

A Tychonoff space X is of countable type if and only if the remainder in any (or some) Hausdorff compactification of X is Lindelof.

Recall that a Tychonoff space X is said to be weakly pseudocompact if there exists a Hausdorff compactification bX such that X is [G.sub.[delta]]-dense in bX, that is, every non-empty [G.sub.[delta]]-set in bX intersects X.

In what follows we shall refer to a

Tychonoff space X satisfying the conditions of the statement of Theorem 8 as a cn-space.

When working with free topological groups, it is also very mportant to know under wh ch cond t ons on a subspace X of a Tychonoff space Y, the subgroup F(X, Y) of F(Y) generated by X is topologically isomorphic to the group F(X), under the natural isomorphism extending the identity embedding of X to Y.

If X is an arbitrary subspace of a Tychonoff space Y, then let [e.sub.X,Y] be the natural embedding mapping from X to Y.

[12,13, Nummela-Pestov] Let X be a dense subspace of a Tychonoff space Y.

Mahmood, "

Tychonoff Spaces in Soft Setting and their Basic Properties," International Journal of Applications of Fuzzy Sets and Artificial Intelligence, vol.

Frink [2] gave a generalised this method to provide Hausdorff compactification for

Tychonoff spaces. In [4] Frink's procedure uses a normal base of closed sets instead of the family of all closed sets as employed by Wallman (6).

[6] Thrivikraman, T., On compactifications of

Tychonoff spaces, Yokohama Math.