1. The notation to be used in this paper agrees with the one which is described in ?1.1 of the following paper On rings of operators. The quotations refer to the bibliography of that paper, quotations from the paper itself will be marked RO. We will make free use of the results of Chapters I and II in RO, excepting however Lemma 2.3.7 in ?2.3. This Lemma is based on the results of the present paper, but it will not be used in the other parts of those sections. Our subject is the introduction of a new topology in the ring of operators B (cf. (18), p. 370) of a space ! (cf. RO, ?1.1, point (b)), which is more appropriate for use under certain conditions than the three topologies of B defined in (18), pp. 378-388. Apart from its independent interest, we need it for application in RO, Lemma 2.3.7 in ?2.3, and in Chapter III. This topology is technically superior to the three above mentioned ones in this: For the "uniform" topology in B the results of our ?4 are true, but those of our ?3 are not. For the "strong" as well as for the "weak" topology in B, ?3 is true, but ?4 is not. (We will give an example for this in ?5.) The "strongest" topology however, as we are going to define it, will have all these properties.