proof of injective images of Baire space
otherwise we may take . In either case, , so it is enough to choose the sets to have diameter no more than with respect to . Note also that any bounded and closed set with respect to this metric is also closed as a subset of .
Let be the condensation points of , which are the points whose neighborhoods all contain uncountably many points of . Then, is a union of countably many countable and open subsets of , so is countable and open. Hence, is uncountable and closed in , and every open subset is uncountable. Choosing any then will not be a closed subset of . So, replacing by if necessary, we may suppose that is not closed as a subset of and, therefore is not compact.
So, for some , cannot be covered by finitely many sets with -diameter no more than (see here (http://planetmath.org/ProofThatAMetricSpaceIsCompactIfAndOnlyIfItIsCompleteAndTotallyBounded)). By separability, there is a sequence of open balls in with diameter less than , and covering . Writing for the -closure of and eliminating any terms such that , then have nonempty interior and hence are uncountable, and
is countable, as required. ∎
Note that is also a difference of closed sets. So, the lemma allows us to inductively choose sets for integers and such that and the following are satisfied.
is closed for all .
is countable and, for , has diameter no more than .
For any we may choose a sequence . Since, for , this set has diameter no more than , then whenever . So, the sequence is Cauchy (http://planetmath.org/CauchySequence) and hence has a limit . Furthermore, as is contained in the closed set
for , then must also be contained in it and hence is in . So
contains and is nonempty. Furthermore, as it has zero diameter, it is a singleton. So is uniquely defined by .
Given any such that for , then and are both in the set , which has diameter no more than . So, and is continuous.
Now let be the countable set
These sets form a basis for the topology on . Clearly,
Choosing any then we can inductively find for such that . Then, setting gives . This shows that
In particular, and, therefore is countable. Finally, as is a difference of closed sets, it is Borel, and equation (1) shows that the inverse of is Borel measurable.