For this purpose, we introduce a natural topology on Milnor's K-groups [K.sup.M.sub.l](k) for a topological field k as the quotient topology induced by the joint determinant map and show that, in case of k = R or C, the natural topology on [K.sup.M.sub.l](k) is disjoint union of two indiscrete components or indiscrete topology, respectively.

For a topological field k, the topology on Milnor's K-group [K.sup.M.sub.l](k) is the quotient topology with respect to the map [[PHI].sub.l] : [Comm.sub.l](k) [right arrow] [K.sup.M.sub.l](k), which is the composite of a natural map [Comm.sub.l](k) [right arrow] [GW.sub.l](k) followed by the group isomorphism [[phi].sub.l] : [GW.sub.l](k) [??] [K.sup.M.sub.l](k) which is described in the proof of Theorem 6.7 of [1].

Since the natural topology on [K.sup.M.sub.l](k) in Definition 1 is the quotient topology, any continuous joint determinant induces a continuous map from [K.sup.M.sub.l](k) into G and vice versa.

Let A be an ([[alpha].sub.n])-galbed algebra with bounded elements, [pi] : A [right arrow] A/I the canonical homomorphism, and O' a neighbourhood of zero in (A/I, [[tau].sub.I]), where [[tau].sub.I] stands for the quotient topology defined by the topology [tau] of A.

On [A.sub.M] we consider the quotient topology [[tau].sub.M] and on [DELTA] the topology [[tau].sub.[DELTA]] := {[[pi].sup.-1](U) : U [member of] [tau]}.