  We show a necessary and sufficient condition for any ordinal number to be a
Polish space. We also prove that for each countable Polish space, there exists
a countable ordinal number that is an upper bound for the first component of
the Cantor-Bendixson characteristic of every compact countable subset of the
aforementioned space. In addition, for any uncountable Polish space, for every
countable ordinal number and for all nonzero natural number, we show the
existence of a compact countable subset of this space such that its
Cantor-Bendixson characteristic equals the previous pair of numbers. Finally,
for each Polish space, we determine the cardinality of the partition, up to
homeomorphisms, of the set of all compact countable subsets of the aforesaid
space.
