  We study properties of some popular topology on the space of Borel
probabilities on a topological ambient space in this paper. We show that the
two types of popular vague topology are equivalent to each other in case the
ambient space is LCH. The two types of setwise topology induced from two
equivalent descriptions of setwisely sequential convergence of probability
measures are also equivalent to each other regardless of the topology on the
ambient space. We give explicit conditions for the two types of vague topology
and the two types of setwise topology to be separable or metrizable on the
space of Borel probabilities. These conditions are either in terms of the
cardinality of the elementary events in the Borel $\sigma$-algebra or some
direct topological assumptions on the ambient space. We give an necessary and
sufficient condition for families of probability measures to be setwisely
relatively compact in case the ambient space is a compact metric space. There
are some extending problems and heuristic schemes on formulating new topologies
on the space of Borel probabilities at the end of the work.

Probability