Let A be a Banach algebra with a bounded approximate identity bounded by 1. Two new topologies τ so and τ wo are introduced on A. We study these topologies and compare them with each other and with the norm topology. The properties of τ so and τ wo are then studied further and we pay attention to the group algebra L 1 (G) of a locally compact group G. Various necessary and sufficient conditions are found for a locally compact group G to be finite.