CONTINUITY, BOUNDEDNESS, CONNECTEDNESS AND THE LINDELOF PROPERTY FOR TOPOLOGICAL-GROUPS

If X is a space and B subset-of-or-equal-to X, we say that B is functionally bounded in X if every continuous real-valued function on X is bounded on B. For a locally compact Abelian group G with character group triple-over-dot, we denote by G+ the underlying group G equipped with the weakest topology which makes every chi epsilon triple-over-dot as above we prove the following: (a) If F subset-of-or-equal-to G then F is Lindelof (functionally bounded) in G if and only if F is Lindelof (functionally bounded) in G+, (b) G is connected (zero-dimensional) if and only if G+ is connected (zero-dimensional), and (c) If G and H are locally compact Abelian groups and empty-set: G --> H is an homomorphism then empty-set: G --> H is continuous if and only if empty-set: G+ --> H+ is continuous. We generalize (c) to a result involving k-spaces.