The exponential function of locally connected compact abelian groups

It is shown that the following four conditions are equivalent for a compact connected abelian group G: (i) the exponential function of G is open onto its image; (ii) G has arbitrarily small connected direct summands N such that G/N is a finite dimensional torus; (iii) the arc component G(a) of the identity is locally arcwise connected; (iv) the character group (G) over cap is a torsion free group in which every finite rank pure subgroup is free and is a direct summand.