Projective topological groups

The notion of a projective topological for a category of topological groups has been defined by Hofmann [4]. Free topological groups have been of interest in the study of topological groups since theirinventionby Markov [5]. It is the purpose of this paper to develop the concept of projective in the category of Abelian topological groups, based upon the requirement that the class of projective topological groups contains the class of Abelian topological groups. Thus the approach used here is necessarily different from that taken by Hofmann [4]. We also will show that the class of projectives so obtained properly contains the class of Abelian topological groups, and will give an example to show that the direct sum of projective topological groups may fail to be projective. NOTATIONS. All topological groups considered are Abelian and Hausdorff. Any unexplained notation or terminology is that of Hewitt and Ross [3]. If X is a completely regular space then A (X) denotes the Abelian topological on X as defined by Markov [5] and A G(X) denotes the Abelian topological on X as defined by Graev [2]. The words free topological group mean a topological as defined by Markov [5]. If B is a topological group, aB denotes the open continuous homomorphism from A (B) onto B defined by aB(b) =b all bEB. If I is an index set and Pi is a topological for each iEI then by EPi (iCI) we mean the direct sum of {P;: iCI} with the relativized product topology. If I= {1, 2, , n}, thenEPi (iEI) iswrittenasP1XP2X * XPn. A sequence of the form f:A-+B-+O is exact if A and B are topological groups and f is a continuous homomorphism from A onto B. DEFINITION 1. A topological G is projective relative to a family F of exact sequences of the form