1G.2] 1G. Borelfunctions and isomorphisms 39

If g(x) = α,then

a(n) = k ⇐⇒ d(x,rk) ≤

2−n−2

&(∀s k)[d(x,rs)

2−n−2].

Thus if

Bn,k = {α : α(n) = k},

each g−1[Bn,k] is a Borel subset of X. It follows that for each basic nbhd N = {α :

α(0) = k0,...,α(n − 1) = kn−1} in N,the set

g−1[N]= g−1[B0,k0

] ∩ ··· ∩

g−1[Bn−1,kn−1

]

is Borel and g is a Borel function.

Now, easily

α ∈ B ⇐⇒ (∀n) d

(

(α),rα(n)

)

≤

2−n−2

&

(

∀k α(n)

)

d

(

(α),rk

)

2−n−2

,

so B is a Π2 0 subset of N. We must refine the construction a bit to get and A with

the same properties, with A a closed set.

Put B in normal form

α ∈ B ⇐⇒ (∀n)(∃s)R(α,n,s),

where R is a clopen pointset by 1B.7 and define A ⊆ N × N by

(α,) ∈ A ⇐⇒ (∀n) R

(

α,n,(n)

)

&

(

∀k (n)

)

¬R(α,n,k) .

Clearly A is closed. Moreover, the projection : N × N → N, (α,) = α takes A

onto B and is one-to-one on A,since

(α,) ∈ A =⇒ (n) = least k such that R(α,n,k).

Hence the composition = ◦ takes A onto X and is continuous, one-to-one.

It is trivial to check that the inverse of

f(x) = g(x),n → least k such that R

(

g(x),n,k

)

isBorel. Theproofiscompletedbycarrying A to N viasometrivialhomeomorphism

of N with N × N, e.g., the map

n0,n1,n2,... →

(

(n0,n2,n4,...),(n1,n3,n5,...)

)

.

The function f of this proof is an example of an interesting class of functions. Let

us temporarily call a function

f : X Y

a good Borel injection if

(1) f is a Borel injection,

(2) there is a Borel surjection

g : Y X

such that g ◦ f is the identity on X, i.e.,

g

(

f(x)

)

= x (x ∈ X).

We refer to any such g as a Borel inverse of f.

It will turn out that every Borel injection is a good Borel injection. This is a special

case of a fairly diﬃcult theorem which we will prove in 2E and again in Chapter 4.

Here we only need show that enough good Borel injections exist.