  A topological space is defined to be banalytic (resp. analytic) if it is the
image of a Polish space under a Borel (resp. continuous) map. A regular
topological space is analytic if and only if it is banalytic and cosmic. Each
(regular) banalytic space has countable spread (and under PFA is hereditarily
Lindel\"of). Applying banalytic spaces to topological groups, we prove that for
a Baire topological group $X$ the following conditions are equivalent: (1) $X$
is Polish, (2) $X$ is analytic, (3) $X$ is banalytic and cosmic, (4) $X$ is
banalytic and has countable pseudocharacter. Under PFA the conditions (1)--(4)
are equivalent to the banalycity of $X$. The conditions (1)--(3) remain
equivalent for any Baire semitopological group.
