Bases for functions beyond the first Baire class
aa r X i v : . [ m a t h . L O ] F e b BASES FOR FUNCTIONS BEYOND THE FIRSTBAIRE CLASS
RAPHA¨EL CARROY AND BENJAMIN D. MILLER
Abstract.
We provide a finite basis for the class of Borel func-tions that are not in the first Baire class, as well as the class ofBorel functions that are not σ -continuous with closed witnesses. Introduction
A topological space is analytic if it is a continuous image of a closedsubset of N N . A subset of a topological space is Borel if it is in the σ -algebra generated by open sets, F σ if it is a union of countably-manyclosed sets, and G δ if it is an intersection of countably-many open sets.Suppose that X and Y are topological spaces. Given a family Γ ofsubsets of X , a function φ : X → Y is Γ -measurable if φ − ( V ) ∈ Γ forevery open set V ⊆ Y . A function is Borel if it is Borel-measurable,
Baire class one if it is F σ -measurable, and σ -continuous with closedwitnesses if its domain is the union of countably-many closed sets onwhich it is continuous. A result of Jayne-Rogers (see [JR82, Theorem1]) ensures that a function from an analytic metric space to a separablemetric space has this property if and only if it is G δ -measurable.A quasi-order on a set Z is a reflexive transitive binary relation ≤ on Z . A set B ⊆ Z is a basis under ≤ for Z if ∀ z ∈ Z ∃ b ∈ B b ≤ z .A closed continuous embedding of φ : X → Y into φ ′ : X ′ → Y ′ consists of a pair of closed continuous embeddings π X : X → X ′ and π Y : φ ( X ) → φ ′ ( X ′ ) such that φ ′ ◦ π X = π Y ◦ φ . Note that the existenceof such a pair depends not only on the graphs of the functions φ and φ ′ , but on Y as well, since different choices of Y ⊇ φ ( X ) can lead todifferent values of φ ( X ). Here we establish the following results. Theorem 1.
There is a twenty-four-element basis under closed contin-uous embeddability for the class of non-Baire-class-one Borel functionsbetween analytic metric spaces.
Mathematics Subject Classification.
Primary 03E15, 26A21, 28A05, 54H05.
Key words and phrases.
Baire class, basis, embedding, sigma-continuous.The authors were supported in part by FWF Grants P28153 and P29999.
Theorem 2.
There is a twenty-seven-element basis under closed con-tinuous embeddability for the class of non- σ -continuous-with-closed-wit-nesses Borel functions between analytic metric spaces. In §
1, we discuss the compactification N ≤ N ∗ of N ≤ N underlying ourarguments, as well as the corresponding compactification N N ∗ of N N .In §
2, we discuss the endomorphisms of N < N underlying our argu-ments. In §
3, we provide a three-element basis for the class of Bairemeasurable functions from N N to separable metric spaces. In §
4, weprovide a three-element basis for the class of non- σ -continuous-with-closed-witnesses Baire-class-one functions from analytic metric spacesto separable metric spaces. In §
5, we provide an eight-element basis forthe class of all functions from N N ∗ \ N N to analytic metric spaces. Andin §
6, we establish Theorems 1 and 2.1.
A compactification of N ≤ N We use s a t to denote the concatenation of sequences s and t , andwe say that s is an initial segment of t , or s ⊑ t , if there exists s ′ forwhich t = s a s ′ . Endow the set N ≤ N ∗ = N ≤ N ∪ { t a ( ∞ ) | t ∈ N < N } with the smallest topology with respect to which the sets of the form { t } and N t = { c ∈ N ≤ N ∗ | t ⊑ c } , where t ∈ N < N , are clopen. Proposition 1.1.
The family B of sets of the form { t } and N t \ ( { t } ∪ S j
The space N ≤ N ∗ is compact.Proof. Suppose, towards a contradiction, that there is an open cover U of N ≤ N ∗ with no finite subcover. Lemma 1.3.
Suppose that t ∈ N < N and no finite set V ⊆ U covers N t .Then there exists j ∈ N such that no finite set V ⊆ U covers N t a ( j ) .Proof. Fix U ∈ U containing t a ( ∞ ). Proposition 1.1 then yields i ∈ N with N t \ ( { t } ∪ S j
By recursively applying Lemma 1.3, we obtain b ∈ N N such that forno i ∈ N is there a finite set V ⊆ U covering N b ↾ i . But Proposition 1.1implies that every open neighborhood of b contains some N b ↾ i .Given a countable set I and a topological space X , we say that asequence ( x i ) i ∈ I ∈ X I converges to a point x ∈ X , or x i → x , if forevery open neighborhood U of x there are only finitely many i ∈ I with x i / ∈ U . When I and X are equipped with partial orders ≤ I and ≤ X ,we say that ( x i ) i ∈ I is decreasing if i ≤ I j = ⇒ x j ≤ X x i for all i, j ∈ I . Proposition 1.4.
The space N ≤ N ∗ has a compatible ultrametric.Proof. Fix a decreasing sequence ( ǫ t ) t ∈ N < N of positive real numbersconverging to zero. Set d ( a, a ) = 0 for all a ∈ N ≤ N ∗ , as well as d ( a, b ) = max { ǫ t | t ∈ { a ↾ min( | a | , i ( a, b )) , b ↾ min( | b | , i ( a, b )) } ∩ N < N } for all distinct a, b ∈ N ≤ N ∗ , where i ( a, b ) = min { i ∈ N | a ↾ i = b ↾ i } .To see that d is an ultrametric, suppose that a, b, c ∈ N ≤ N ∗ arepairwise distinct. Observe that if i ( a, c ) < max { i ( a, b ) , i ( b, c ) } , then d ( a, c ) ∈ { d ( b, c ) , d ( a, b ) } , so d ( a, c ) ≤ max { d ( a, b ) , d ( b, c ) } . And if i ( a, c ) = max { i ( a, b ) , i ( b, c ) } , then setting i = i ( a, b ) = i ( a, c ) = i ( b, c ),it follows that d ( a, c ) = max { ǫ t | t ∈ { a ↾ i, c ↾ i } ∩ N < N }≤ max { ǫ t | t ∈ { a ↾ i, b ↾ i, c ↾ i } ∩ N < N } = max { d ( a, b ) , d ( b, c ) } . And if i ( a, c ) > max { i ( a, b ) , i ( b, c ) } , then setting ǫ = d ( a, b ) = d ( b, c )and t = a ↾ i ( a, b ) = c ↾ i ( b, c ), it follows that d ( a, c ) ≤ ǫ t ≤ ǫ , andtherefore d ( a, c ) ≤ max { d ( a, b ) , d ( b, c ) } .As { t } = B ( t, ǫ t ) and N t \ { t } = B ( N t \ { t } , ǫ t ) for all t ∈ N < N , and N t \ ( { t }∪ S j ≤ i N t a ( j ) ) = B ( N t \ ( { t }∪ S j ≤ i N t a ( j ) ) , min( { ǫ t a ( j ) | j ≤ i } ))for all i ∈ N and t ∈ N < N , Proposition 1.1 ensures that every opensubset of N ≤ N ∗ is d -open.Given b ∈ N N and ǫ >
0, fix i ∈ N with ǫ b ↾ i < ǫ , set t = b ↾ i , and notethat N t ⊆ B ( b, ǫ ). Given t ∈ N < N and ǫ >
0, fix i ∈ N with ǫ t a ( j ) < ǫ for all j ≥ i , and observe that N t \ ( { t } ∪ S j
The meet of sequences s, t ∈ N < N is the sequence r = s ∧ t of maximallength for which r ⊑ s and r ⊑ t . A ∧ -embedding is an injection π : N < N → N < N such that π ( s ∧ t ) = π ( s ) ∧ π ( t ) for all s, t ∈ N < N . Proposition 2.1.
Suppose that π : N < N → N < N . Then π is a ∧ -embedding if and only if the following conditions hold: (1) ∀ i ∈ N ∀ t ∈ N < N π ( t ) ⊏ π ( t a ( i )) . (2) ∀ i, j ∈ N ∀ t ∈ N < N ( i = j = ⇒ π ( t a ( i ))( | π ( t ) | ) = π ( t a ( j ))( | π ( t ) | )) .Proof. Suppose first that π is a ∧ -embedding. To see that condition (1)holds, observe that if i ∈ N and t ∈ N < N , then π ( t ) = π ( t ) ∧ π ( t a ( i )),so π ( t ) ⊑ π ( t a ( i )), thus π ( t ) ⊏ π ( t a ( i )). And to see that condition(2) holds, observe that if i, j ∈ N are distinct and t ∈ N < N , then π ( t ) = π ( t a ( i )) ∧ π ( t a ( j )), so π ( t a ( i ))( | π ( t ) | ) = π ( t a ( j ))( | π ( t ) | ).Suppose now that π satisfies conditions (1) and (2). To see that π is a ∧ -embedding, suppose that s, t ∈ N < N are distinct, and define r = s ∧ t . By reversing the roles of s and t if necessary, we can assumethat | s | > | r | , so π ( r ) ⊏ π ( s ), thus either r = t or ( | t | > | r | and π ( s )( | π ( r ) | ) = π ( t )( | π ( r ) | )). In both cases, it follows that π ( s ) = π ( t )and π ( r ) = π ( s ) ∧ π ( t ). Remark 2.2.
In particular, it follows that if π : N < N → N < N has theproperty that π ( t ) a ( i ) ⊑ π ( t a ( i )) for all i ∈ N and t ∈ N < N , then π is a ∧ -embedding.There is a simple but useful means of amalgamating appropriatelyindexed families of ∧ -embeddings. Proposition 2.3.
Suppose that ( π t ) t ∈ N < N is a sequence of ∧ -embeddingswith the property that π t ( N < N ) ⊆ N t for all t ∈ N < N . Then the function π : N < N → N < N given by π ( t ) = ( Q n ≤| t | π t ↾ n )( t ) is also a ∧ -embedding.Proof. Note that if i ∈ N and t ∈ N < N , then t a ( i ) ⊑ π t a ( i ) ( t a ( i )),so Proposition 2.1 ensures that ( Q n ≤| t | π t ↾ n )( t a ( i )) ⊑ π ( t a ( i )),thus π ( t ) ⊏ ( Q n ≤| t | π t ↾ n )( t a ( i )) ⊑ π ( t a ( i )). It also implies that if i = j , then ( Q n ≤| t | π t ↾ n )( t a ( i ))( | π ( t ) | ) = ( Q n ≤| t | π t ↾ n )( t a ( j ))( | π ( t ) | ),so π ( t a ( i ))( | π ( t ) | ) = π ( t a ( j ))( | π ( t ) | ). One last application ofProposition 2.1 therefore ensures that π is a ∧ -embedding.We next consider the connection between ∧ -embeddings and closedcontinuous embeddings. HE FIRST BAIRE CLASS 5
Proposition 2.4.
Every ∧ -embedding π : N < N → N < N has a uniqueextension to a (necessarily injective) continuous map π : N ≤ N ∗ → N ≤ N ∗ ,given by π ( b ) = S i ∈ N π ( b ↾ i ) and π ( t a ( ∞ )) = π ( t ) a ( ∞ ) for all b ∈ N N and t ∈ N < N .Proof. Suppose that π : N ≤ N ∗ → N ≤ N ∗ is a continuous extension of π . If b ∈ N N , then b ↾ i → b , and since ( π ( b ↾ i )) i ∈ N is strictly increasing byProposition 2.1, it follows that π ( b ) = S i ∈ N π ( b ↾ i ). If t ∈ N < N , then t a ( i ) → t a ( ∞ ), and since π ( t ) = π ( t a ( i )) ∧ π ( t a ( j )) for alldistinct i, j ∈ N , it follows that π ( t a ( ∞ )) = π ( t ) a ( ∞ ).To see that these constraints actually define a continuous function,note that if t ∈ N < N , then either π − ( N t ) = ∅ or there exists s ∈ N < N of minimal length with t ⊑ π ( s ), in which case π − ( N t ) = N s .To see that π is injective, it is enough to check that its restriction to N N is injective. Towards this end, suppose that a, b ∈ N N are distinct,fix i ∈ N least for which a ( i ) = b ( i ), set t = a ↾ i = b ↾ i , and observethat π ( t a ( a ( i )))( | π ( t ) | ) = π ( t a ( b ( i )))( | π ( t ) | ) by Proposition 2.1,thus π ( a ) and π ( b ) are distinct. Remark 2.5.
It follows that the extension associated with the com-position of two ∧ -embeddings is the composition of their extensions.Given a function φ : X → Y and sets X ′ ⊆ X and Y ′ ⊇ φ ( X ′ ), let φ ↾ X ′ → Y ′ denote the function ψ : X ′ → Y ′ given by φ ( x ) = ψ ( x ) forall x ∈ X ′ . Compactness ensures that if π is a ∧ -embedding, then π and π ↾ N N ∗ are closed continuous embeddings. The following observationsshow that so too are π ↾ N N → N N and π ↾ N N ∗ \ N N → N N ∗ \ N N . Proposition 2.6.
Suppose that π : N < N → N < N is a ∧ -embedding.Then π ↾ N N → N N is closed.Proof. It is sufficient to show that every sequence ( b n ) n ∈ N of elements of N N for which ( π ( b n )) n ∈ N converges to an element of N N is itself conver-gent to an element of N N . As ( π ( b n ) ↾ i ) n ∈ N is eventually constant forall i ∈ N , a simple induction shows that ( b n ↾ i ) n ∈ N is also eventuallyconstant for all i ∈ N , so ( b n ) n ∈ N converges to an element of N N . Proposition 2.7.
Suppose that π : N < N → N < N is a ∧ -embedding.Then π ↾ N N ∗ \ N N → N N ∗ \ N N is closed.Proof. It is sufficient to show that every sequence ( s n ) n ∈ N of elementsof N < N such that ( π ( s n )) n ∈ N converges to t a ( ∞ ) for some t ∈ N < N has a subsequence converging to an element of N N ∗ \ N N . By passing toa subsequence, we can assume that π ( s m ) ∧ π ( s n ) = t for all distinct m, n ∈ N . Let s be the ⊑ -minimal element of N < N for which t ⊑ π ( s ).Then s m ∧ s n = s for all distinct m, n ∈ N , thus s n → s a ( ∞ ). R. CARROY AND B.D. MILLER
A set T ⊆ N < N is ⊑ -dense if ∀ s ∈ N < N ∃ t ∈ T s ⊑ t . More generally,a set T ⊆ N < N is ⊑ -dense below r ∈ N < N if ∀ s ∈ N < N ∃ t ∈ T r a s ⊑ t . Proposition 2.8.
Suppose that T ⊆ N < N . Then there is a ∧ -embedding π : N < N → N < N such that π ( N < N ) ⊆ T or π ( N < N ) ⊆ ∼ T .Proof. Fix S ∈ { T, ∼ T } which is ⊑ -dense below some s ∈ N < N , andrecursively construct a function π : N < N → N s ∩ S with the propertythat π ( t ) a ( i ) ⊑ π ( t a ( i )) for all i ∈ N and t ∈ N < N . Proposition 2.9.
Suppose that C ⊆ N N is a non-meager set with theBaire property. Then there is a ∧ -embedding π : N < N → N < N with theproperty that π ( N N ) ⊆ C .Proof. Fix s ∈ N < N for which C is comeager in N s ∩ N N , as well asdense open sets U n ⊆ N s ∩ N N with the property that T n ∈ N U n ⊆ C .Set T n = { t ∈ N < N | N t ∩ N N ⊆ U n } for all n ∈ N , and recursivelyconstruct a function π : N < N → N s ∩ N < N such that π ( N n ) ⊆ T n for all n ∈ N and π ( t ) a ( i ) ⊑ π ( t a ( i )) for all i ∈ N and t ∈ N < N .3. Baire measurable functions on N N Here we provide a basis for the class of Baire measurable functionsfrom N N to separable metric spaces. Proposition 3.1.
Suppose that X is a second countable topologicalspace and φ : N N → X is Baire measurable. Then there is a ∧ -embedding π : N < N → N < N for which φ ◦ π is continuous.Proof. Fix a comeager set C ⊆ N N on which φ is continuous, and appealto Proposition 2.9 to obtain a ∧ -embedding π : N < N → N < N with theproperty that π ( N N ) ⊆ C . Proposition 3.2.
Suppose that X is a metric space and φ : N N → X is continuous. Then there is a ∧ -embedding π : N < N → N < N with theproperty that diam φ ( N π ( t ) ) → .Proof. Fix a sequence ( ǫ t ) t ∈ N < N of positive real numbers converging tozero, note that the continuity of φ ensures that for all t ∈ N < N theset T t = { s ∈ N < N | diam φ ( N s ) < ǫ t } is ⊑ -dense, and recursivelyconstruct a function π : N < N → N < N such that π ( t ) ∈ T t for all t ∈ N < N and π ( t ) a ( i ) ⊑ π ( t a ( i )) for all i ∈ N and t ∈ N < N .Given a countable set I and a topological space X , we say that asequence ( X i ) i ∈ I of subsets of X converges to a point x ∈ X , or X i → x ,if for every open neighborhood U of x , all but finitely many i ∈ I havethe property that X i ⊆ U . We say that ( X i ) i ∈ I is discrete if for all HE FIRST BAIRE CLASS 7 x ∈ X there is an open neighborhood U of x such that all but finitelymany i ∈ I have the property that U ∩ X i = ∅ . Proposition 3.3.
Suppose that X is a metric space and φ : N N → X has the property that diam φ ( N t a ( i ) ) → for all t ∈ N < N . Then there isa ∧ -embedding π : N < N → N < N such that ( φ ( N π ( t a ( i )) )) i ∈ N is convergentor discrete for all t ∈ N < N .Proof. For each t ∈ N < N , the fact that diam φ ( N t a ( i ) ) → ι t : N → N for which ( φ ( N t a ( ι t ( i )) )) i ∈ N is convergentor discrete. Define π : N < N → N < N by choosing π ( ∅ ) ∈ N < N arbitrarilyand setting π ( t a ( i )) = π ( t ) a ( ι π ( t ) ( i )) for all i ∈ N and t ∈ N < N .We say that a function φ : X → Y is nowhere constant if there is nonon-empty open set U ⊆ X on which φ is constant. Proposition 3.4.
Suppose that X is a metric space and φ : N N → X is continuous and nowhere constant. Then there is a ∧ -embedding π : N < N → N < N such that ∀ i ∈ N ∀ t ∈ N < N φ ( N π ( t a ( i )) ) ∩ S j ∈ N \{ i } φ ( N π ( t a ( j )) ) = ∅ . Proof.
Clearly each φ ( N t ) is infinite. Lemma 3.5.
For all t ∈ N < N , there is a function ι t : N → N < N \ {∅} such that ( ι t ( i )(0)) i ∈ N is injective and the closures of φ ( N t a ι t ( i ) ) and S j ∈ N \{ i } φ ( N t a ι t ( j ) ) are disjoint for all i ∈ N .Proof. As each φ ( N t a ( i ) ) is infinite, there are extensions b i ∈ N N of t a ( i ) such that φ ( b i ) / ∈ { φ ( b j ) | j < i } for all i ∈ N . Fix a subsequence( a i ) i ∈ N of ( b i ) i ∈ N for which { φ ( a i ) | i ∈ N } is discrete. For each i ∈ N ,fix ǫ i > φ ( a j ) / ∈ B ( φ ( a i ) , ǫ i ) for all j ∈ N \ { i } , as well as ι t ( i ) ∈ N < N \ {∅} with t a ι t ( i ) ⊑ a i and φ ( N t a ι t ( i ) ) ⊆ B ( φ ( a i ) , ǫ i / i ∈ N for whichsome x ∈ X is in the closures of φ ( N t a ι t ( i ) ) and S j ∈ N \{ i } φ ( N t a ι t ( j ) ).Then there exist j ∈ N \ { i } and y ∈ φ ( N t a ι t ( j ) ) with the property that d ( x, y ) ≤ ǫ i /
3, in which case d ( φ ( a i ) , φ ( a j )) ≤ d ( φ ( a i ) , x ) + d ( x, y ) + d ( y, φ ( a j )) < ǫ i / ǫ i / ǫ j / ≤ max { ǫ i , ǫ j } , so φ ( a i ) ∈ B ( φ ( a j ) , ǫ j ) or φ ( a j ) ∈ B ( φ ( a i ) , ǫ i ), a contradiction.Define π : N < N → N < N by choosing π ( ∅ ) ∈ N < N arbitrarily and set-ting π ( t a ( i )) = π ( t ) a ι π ( t ) ( i ) for all i ∈ N and t ∈ N < N . R. CARROY AND B.D. MILLER
We now obtain our main result stabilizing the topological behaviorof Baire measurable functions from N N to separable metric spaces. Theorem 3.6.
Suppose that X is a separable metric space and φ : N N → X is Baire measurable. Then there is a ∧ -embedding π : N < N → N < N such that φ ◦ π is constant or extends to a closed continuous embeddingon N N or N N ∗ .Proof. By Remark 2.5, we are free to replace φ by its composition withthe extension of any ∧ -embedding. For example, by Proposition 3.1,we can assume that φ is continuous.If there exists s ∈ N < N for which φ ↾ N s is constant, then define π : N < N → N < N by π ( t ) = s a t for all t ∈ N < N , so φ ◦ π is constant.Otherwise, Propositions 2.8, 3.2, 3.3, and 3.4 yield a ∧ -embedding π : N < N → N < N such that diam φ ( N π ( t ) ) →
0, ( φ ( N π ( t a ( i )) )) i ∈ N is con-vergent for all t ∈ N < N or discrete for all t ∈ N < N , and ∀ i ∈ N ∀ t ∈ N < N φ ( N π ( t a ( i )) ) ∩ S j ∈ N \{ i } φ ( N π ( t a ( j )) ) = ∅ . As π ( N t ) ⊆ N π ( t ) for all t ∈ N < N , it follows that ∀ i ∈ N ∀ t ∈ N < N ( φ ◦ π )( N t a ( i ) ) ∩ S j ∈ N \{ i } ( φ ◦ π )( N t a ( j ) ) = ∅ . So by replacing φ with φ ◦ π , we can assume that diam φ ( N t ) → φ ( N t a ( i ) )) i ∈ N is convergent for all t ∈ N < N or discrete for all t ∈ N < N ,and( † ) ∀ i ∈ N ∀ t ∈ N < N φ ( N t a ( i ) ) ∩ S j ∈ N \{ i } φ ( N t a ( j ) ) = ∅ . To see that φ is injective, note that if a, b ∈ N N are distinct, thenthere is a least i ∈ N for which a ( i ) = b ( i ). Setting t = a ↾ i = b ↾ i , itfollows from ( † ) that φ ( N t a ( a ( i )) ) and φ ( N t a ( b ( i )) ) are disjoint, thus φ ( a )and φ ( b ) are distinct.We next check that if ( φ ( N t a ( i ) )) i ∈ N is discrete for all t ∈ N < N , then φ is a closed continuous embedding. It is sufficient to show that everysequence ( b n ) n ∈ N of elements of N N for which ( φ ( b n )) n ∈ N converges tosome x ∈ X is itself convergent. But a straightforward recursive argu-ment yields b ∈ N N such that x is in the closure of φ ( N b ↾ i ) for all i ∈ N ,so ( † ) ensures that x is not in the closure of S j ∈ N \{ b ( i ) } φ ( N b ↾ i a ( j ) ) forall i ∈ N , thus ( b n ↾ i ) n ∈ N is eventually constant with value b ↾ i for all i ∈ N , hence b n → b .It remains to check that if ( φ ( N t a ( i ) )) i ∈ N is convergent for all t ∈ N < N ,then the extension of φ to N N ∗ given by φ ( t a ( ∞ )) = lim i →∞ φ ( N t a ( i ) )for all t ∈ N < N is a closed continuous embedding. To see that φ isinjective, note that if c, d ∈ N N ∗ are distinct, then there is a least i ∈ N HE FIRST BAIRE CLASS 9 with c ( i ) = d ( i ). By reversing the roles of c and d if necessary, wecan assume that c ( i ) = ∞ . Set t = c ↾ i = d ↾ i , and appeal to( † ) to see that φ ( c ) is in the closure of φ ( N t a ( c ( i )) ) but φ ( d ) is not, so φ ( c ) = φ ( d ). To see that φ is continuous, suppose that c ∈ N N ∗ and U is an open neighborhood of φ ( c ), and fix an open neighborhood V of φ ( c ) whose closure is contained in U . If c ∈ N N , then there exists i ∈ N for which φ ( N c ↾ i ) ⊆ V , thus N c ↾ i is an open neighborhood of c whoseimage under φ is contained in U . Otherwise, there exists t ∈ N < N forwhich c = t a ( ∞ ), as well as i ∈ N for which φ ( N t \ S j
Suppose that X is a separable metric space, φ : N N → X is Baire measurable, π : N < N → N < N is a ∧ -embedding, and φ ◦ π is constant or extends to a closed continuous embedding on N N or N N ∗ . Then there exist φ ∈ { c N N } ∪ { ι N N ,Z | Z ∈ { N N , N N ∗ }} and ψ : φ ( N N ) → φ ( N N ) with the property that ( π ↾ N N → N N , ψ ) is aclosed continuous embedding of φ into φ .Proof. If φ ◦ π is constant, then set φ = c N N and let ψ be the uniquefunction from c N N ( N N ) to ( φ ◦ π )( N N ). If φ ◦ π extends to a closedcontinuous embedding ψ on Z ∈ { N N , N N ∗ } , then set φ = ι N N ,Z .4. Baire-class-one functions that are not σ -continuouswith closed witnesses Here we strengthen [Sol98, Theorem 3.1] by providing a basis for theclass of non- σ -continuous-with-closed-witnesses Baire-class-one func-tions from analytic metric spaces to separable metric spaces. Proposition 4.1.
Suppose that X is a metric space and φ : N N ∗ → X has the property that φ ↾ N N is continuous. Then there is a ∧ -embedding π : N < N → N < N such that either ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ or φ ◦ π is continuous at every point of N N .Proof. We can assume that there is no s ∈ N < N with the property thatinf { d ( φ ( s a b ) , φ ( s a t a ( ∞ ))) | b ∈ N N and t ∈ N < N } >
0, sinceotherwise the ∧ -embedding π : N < N → N < N given by π ( t ) = s a t forall t ∈ N < N has the property that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ . Lemma 4.2.
Suppose that ǫ > and s ∈ N < N . Then there exists t ∈ N < N with d ( φ ( s a t a b ) , φ ( s a t a ( ∞ ))) < ǫ for all b ∈ N N .Proof. Fix δ < ǫ and u ∈ N < N with diam φ ( N s a u ∩ N N ) < δ , and b ∈ N N and v ∈ N < N with d ( φ ( s a u a b ) , φ ( s a u a v a ( ∞ ))) < ǫ − δ , andset t = u a v .Fix a sequence ( ǫ n ) n ∈ N of positive real numbers converging to zero,and recursively construct a function π : N < N → N < N with the propertythat d ( φ ( π ( t ) a b ) , φ ( π ( t ) a ( ∞ ))) < ǫ | t | for all b ∈ N N and t ∈ N < N ,and π ( t ) a ( i ) ⊑ π ( t a ( i )) for all i ∈ N and t ∈ N < N .We say that a metric space is ǫ -discrete if all distinct points havedistance at least ǫ from one another. Proposition 4.3.
Suppose that X is a metric space, φ : N N ∗ \ N N → X , ǫ > , and t ∈ N < N . Then there is a ∧ -embedding π : N < N → N t ∩ N < N with the property that φ ◦ π is an injection into an ǫ -discrete set or ( φ ◦ π )( N N ∗ \ N N ) is contained in the ǫ -ball around a point of φ ( N t ) .Proof. If for no finite set F ⊆ φ ( N N ∗ \ N N ) and extension u of t is itthe case that φ ( N u ) ⊆ B ( F, ǫ ), then fix an enumeration ( t n ) n ∈ N of N < N with the property that t m ⊑ t n = ⇒ m ≤ n for all m, n ∈ N , andrecursively construct π : N < N → N t ∩ N < N such that φ ( π ( t n ) a ( ∞ )) / ∈B ( { φ ( π ( t m ) a ( ∞ )) | m < n } , ǫ ) and π ( t ′ n ) a ( n ) ⊑ π ( t n ) for all n > t ′ n is the maximal proper initial segment of t n .Otherwise, there exists x ∈ φ ( N N ∗ \ N N ) with the property that theset S = { s ∈ N < N | φ ( s a ( ∞ )) ∈ B ( x, ǫ ) } is ⊑ -dense below someextension u of t , in which case we can recursively construct a function π : N < N → N u ∩ S with the property that π ( v ) a ( i ) ⊑ π ( v a ( i )) forall i ∈ N and v ∈ N < N . Proposition 4.4.
Suppose that X is a metric space and φ : N N ∗ \ N N → X . Then there is a ∧ -embedding π : N < N → N < N such that φ ◦ π is aninjection into an ǫ -discrete set for some ǫ > or diam ( φ ◦ π )( N t ) → .Proof. Suppose that for no ǫ > ∧ -embedding π : N < N → N < N such that φ ◦ π is an injection into an ǫ -discrete set, fix a sequence( ǫ t ) t ∈ N < N of positive real numbers converging to zero, and recursivelyapply Proposition 4.3 to the functions φ t = φ ◦ Q n< | t | π t ↾ n to obtain ∧ -embeddings π t : N < N → N t ∩ N < N such that ( φ ◦ Q n ≤| t | π t ↾ n )( N N ∗ \ N N )is contained in an ǫ t -ball for all t ∈ N < N . Let π be the ∧ -embeddingobtained from applying Proposition 2.3 to ( π t ) t ∈ N < N , and observe thatdiam ( φ ◦ π )( N t ) → HE FIRST BAIRE CLASS 11
Define p : N N ∗ \ N N → N < N by setting p ( t a ( ∞ )) = t for all t ∈ N < N .Let N < N ∗ = N < N ∪ {∞} denote the one-point compactification of N < N . Theorem 4.5.
Suppose that X is an analytic metric space, Y is aseparable metric space, and φ : X → Y is a Baire-class-one functionthat is not σ -continuous with closed witnesses. Then there exists φ ∈{ c N N } ∪ { ι N N ,Z | Z ∈ { N N , N N ∗ }} for which there is a closed continuousembedding of φ ∪ p into φ .Proof. By the Jayne-Rogers theorem (see, for example, [JR82, The-orem 1]), we can assume that φ is not G δ -measurable. Hurewicz’sdichotomy theorem for F σ sets then yields a closed continuous embed-ding ψ : N N ∗ → X with ( φ ◦ ψ )( N N ) ∩ ( φ ◦ ψ )( N N ∗ \ N N ) = ∅ (see, forexample, [CMS, Theorem 4.2]). As ( ψ, id ( φ ◦ ψ )( N N ∗ ) ) is a closed continu-ous embedding of φ ◦ ψ into φ , by replacing the latter with the former,we can assume that X = N N ∗ and φ ( N N ) ∩ φ ( N N ∗ \ N N ) = ∅ .By Proposition 3.1, there is a ∧ -embedding π : N < N → N < N for which( φ ◦ π ) ↾ N N is continuous. By composing π with the ∧ -embedding givenby Proposition 4.1, we can assume that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ or φ ◦ π is continuous at every point of N N . As φ is Baire class one,the former possibility would imply that the pre-images of ( φ ◦ π )( N N )and ( φ ◦ π )( N N ∗ \ N N ) under φ ◦ π are disjoint dense G δ subsets of N N ∗ ,so the latter holds. By Proposition 4.4, we can assume that eitherthere exists ǫ > φ ◦ π ) ↾ N N ∗ \ N N is an injection into an ǫ -discrete set, or diam ( φ ◦ π )( N t ∩ ( N N ∗ \ N N )) →
0. As the formerpossibility contradicts the facts that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ and ( φ ◦ π )( N N ) ⊆ ( φ ◦ π )( N N ∗ \ N N ), it follows that the latter holds. Byapplying Proposition 4.3 with any ǫ > t ∈ N < N , but replacingthe given metric on X by one with respect to which all pairs of distinctpoints have distance at least ǫ from one another, we can assume that( φ ◦ π ) ↾ N N ∗ \ N N is either constant or injective. Lemma 4.6.
Suppose that ( s n ) n ∈ N is an injective sequence of elementsof N < N and ( b n ) n ∈ N is a sequence of elements of N N such that s n ⊑ b n for all n ∈ N . Then d X (( φ ◦ π )( b n ) , ( φ ◦ π )( s n a ( ∞ ))) → .Proof. Simply note that ( φ ◦ π )( b n ) ∈ ( φ ◦ π )( N s n ∩ ( N N ∗ \ N N )) for all n ∈ N and diam ( φ ◦ π )( N s n ∩ ( N N ∗ \ N N )) → φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ and( φ ◦ π )( N N ) ⊆ ( φ ◦ π )( N N ∗ \ N N ), Lemma 4.6 ensures that ( φ ◦ π ) ↾ N N ∗ \ N N is not constant, and is therefore injective. Along with thefact that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ , Lemma 4.6 ensures that( φ ◦ π )( N N ∗ \ N N ) is discrete. By Theorem 3.6, we can assume that ( φ ◦ π ) ↾ N N is constant orextends to a closed continuous embedding on N N or N N ∗ .We will now complete the proof by showing that there exist φ ∈{ c N N } ∪ { ι N N ,Z | Z ∈ { N N , N N ∗ }} and ψ : φ ( N N ∗ ) ∪ N < N → φ ( X ) forwhich ( π ↾ N N ∗ → N N ∗ , ψ ) is a closed continuous embedding of φ ∪ p into φ .If ( φ ◦ π ) ↾ N N is constant with value y ∈ Y , then set φ = c N N , andnote that the extension ψ of φ ◦ π ◦ p − to N < N ∗ given by ψ ( ∞ ) = y is injective. As Lemma 4.6 ensures that ( φ ◦ π )( s n a ( ∞ )) → y forevery injective sequence ( s n ) n ∈ N of elements of N < N , it follows that ψ is continuous, so the compactness of N < N ∗ ensures that ψ is a closedcontinuous embedding.If ( φ ◦ π ) ↾ N N is a closed continuous embedding, then set φ = ι N N , N N , and note that the extension ψ of φ ◦ π ◦ p − to N ≤ N given by ψ ↾ N N = ( φ ◦ π ) ↾ N N is a continuous injection. To see that it isclosed, it is enough to show that every injective sequence ( a n ) n ∈ N ofelements of N ≤ N for which ( ψ ( a n )) n ∈ N converges to some point y ∈ Y has a subsequence converging to a point of N N . As N ≤ N ∗ is compact,by passing to a subsequence, we can assume that ( a n ) n ∈ N convergesto a point of N ≤ N ∗ . As every point of N < N is isolated, it thereforeconverges to a point of N N ∗ . And if there exists t ∈ N < N for which a n → t a ( ∞ ), then there are extensions b n ∈ N N of a n for all n ∈ N ,in which case b n → t a ( ∞ ) and ψ ( b n ) → y by Lemma 4.6. Fix n ∈ N sufficiently large that ( φ ◦ π )( b m ) = y for all m ≥ n , and observe that { b m | m ≥ n } is a closed subset of N N whose image under φ ◦ π is notclosed, contradicting the fact that ( φ ◦ π ) ↾ N N is closed.If ( φ ◦ π ) ↾ N N extends to a closed continuous embedding ψ ′ on N N ∗ ,then set φ = ι N N , N N ∗ , and note that the extension ψ of φ ◦ π ◦ p − to N ≤ N ∗ given by ψ ↾ N N ∗ = ψ ′ ↾ N N ∗ is injective. To see that it iscontinuous, suppose that ( t n ) n ∈ N is an injective sequence of elementsof N < N converging to t a ( ∞ ) for some t ∈ N < N , fix b n ∈ N t n ∩ N N forall n ∈ N , and observe that the continuity of ψ ′ ensures that ψ ( b n ) → ψ ( t a ( ∞ )), thus Lemma 4.6 implies that ψ ( t n ) → ψ ( t a ( ∞ )). As N ≤ N ∗ is compact, it follows that ψ is a closed continuous embedding.5. Functions on N N ∗ \ N N Here we provide a basis for the class of all functions from N N ∗ \ N N to analytic metric spaces. Proposition 5.1.
Suppose that X is a topological space, φ : N N ∗ \ N N → X is injective, and x ∈ X . Then there is a ∧ -embedding π : N < N → N < N such that x / ∈ ( φ ◦ π )( N N ∗ \ N N ) . HE FIRST BAIRE CLASS 13
Proof.
Fix s ∈ N < N such that x / ∈ φ ( N s ), and define π : N < N → N < N by π ( t ) = s a t for all t ∈ N < N . Proposition 5.2.
Suppose that X is a metric space and φ : N N ∗ \ N N → X . Then there is a ∧ -embedding π : N < N → N < N with the property that (( φ ◦ π )( t a ( i, ∞ ))) i ∈ N is convergent or { ( φ ◦ π )( t a ( i, ∞ )) | i ∈ N } is closed and discrete for all t ∈ N < N .Proof. For each t ∈ N < N , there is an injection ι t : N → N for which( φ ( t a ( ι t ( i ) , ∞ ))) i ∈ N is convergent or { φ ( t a ( ι t ( i ) , ∞ )) | i ∈ N } isclosed and discrete. Define π : N < N → N < N by choosing π ( ∅ ) ∈ N < N arbitrarily and setting π ( t a ( i )) = π ( t ) a ( ι π ( t ) ( i )) for all i ∈ N and t ∈ N < N , and note that ( φ ◦ π )( t a ( i, ∞ )) = φ ( π ( t a ( i )) a ( ∞ )) = φ ( π ( t ) a ( ι π ( t ) ( i ) , ∞ )) for all i ∈ N and t ∈ N N . Proposition 5.3.
Suppose that X is a metric space, φ : N N ∗ \ N N → X , F ⊆ X is finite, and t ∈ N < N . Then there is a ∧ -embedding π : N < N →N t ∩ N < N such that either (( φ ◦ π )( u a ( ∞ ))) u ∈ N < N converges to anelement of F or the closure of ( φ ◦ π )( N N ∗ \ N N ) is disjoint from F .Proof. If the set S ǫ = { s ∈ N < N | φ ( s a ( ∞ )) ∈ B ( F, ǫ ) } is ⊑ -densebelow t for all ǫ >
0, then there exist an extension u of t and x ∈ F such that the set S ǫ,x = { s ∈ N < N | φ ( s a ( ∞ )) ∈ B ( x, ǫ ) } is ⊑ -dense below u for all ǫ >
0. Fix a sequence ( ǫ v ) v ∈ N < N of positivereal numbers converging to zero, and recursively construct a function π : N < N → N u ∩ N < N such that π ( v ) ∈ S ǫ v ,x for all v ∈ N < N and π ( v ) a ( i ) ⊑ π ( v a ( i )) for all i ∈ N and v ∈ N < N , and observe that( φ ◦ π )( v a ( ∞ )) → x .Otherwise, fix ǫ > u of t with the property that N u ∩ S ǫ = ∅ , define π : N < N → N u ∩ N < N by π ( v ) = u a v , and notethat the closure of ( φ ◦ π )( N N ∗ \ N N ) is disjoint from F .For the rest of this section, it will be convenient to fix an enumeration( t n ) n ∈ N of N < N such that t m ⊑ t n = ⇒ m ≤ n for all m, n ∈ N . Proposition 5.4.
Suppose that X is a metric space and φ : N N ∗ \ N N → X . Then there is a ∧ -embedding π : N < N → N < N with the property that (( φ ◦ π )( t a ( ∞ ))) t ∈ N < N converges or for no natural numbers m < n is ( φ ◦ π )( t m a ( ∞ )) or a limit point of { ( φ ◦ π )( t m a ( i, ∞ )) | i ∈ N } inthe closure of ( φ ◦ π )( N t n ) .Proof. Suppose that for no ∧ -embedding π : N < N → N < N is the se-quence (( φ ◦ π )( t a ( ∞ ))) t ∈ N < N convergent. By Proposition 5.2, we canassume that ( φ ( t a ( i, ∞ ))) i ∈ N is convergent or { φ ( t a ( i, ∞ )) | i ∈ N } is closed and discrete for all t ∈ N < N . By recursively applying Lemma φ t = φ ◦ Q k< | t | π t ↾ k , we obtain ∧ -embeddings π t : N < N → N t ∩ N < N such that for no natural numbers m < n is ( φ ◦ Q k ≤| t m | π t m ↾ k )( t m a ( ∞ )) or a limit point of { ( φ ◦ Q k ≤| t m | π t m ↾ k )( t m a ( i, ∞ )) | i ∈ N } in the closure of ( φ ◦ Q k ≤| t n | π t n ↾ k )( N t n ). Let π bethe ∧ -embedding obtained from applying Proposition 2.3 to ( π t ) t ∈ N < N ,and observe that for no natural numbers m < n is it the case that( φ ◦ π )( t m a ( ∞ )) or a limit point of { ( φ ◦ π )( t m a ( i, ∞ )) | i ∈ N } inthe closure of ( φ ◦ π )( N t n ). Theorem 5.5.
Suppose that X is an analytic metric space and φ : N N ∗ \ N N → X . Then there is a ∧ -embedding π : N < N → N < N such that φ ◦ π is constant, φ ◦ π extends to a closed continuous embedding on N N ∗ \ N N or N N ∗ , or φ ◦ π ◦ p − extends to a closed continuous embedding on N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , or N ≤ N ∗ .Proof. As before, we will repeatedly precompose φ with appropriate ∧ -embeddings, albeit this time so as to stabilize the behavior of thefunction ψ = φ ◦ p − , as opposed to that of the function φ itself. Byapplying Proposition 4.3 with any ǫ > t ∈ N < N , but replacingthe given metric on X by one with respect to which all pairs of distinctpoints have distance at least ǫ from one another, we can assume that ψ is either constant or injective. As φ is constant in the former case,we can assume that we are in the latter.By Proposition 4.4, we can ensure that ψ ( N < N ) is closed and discreteor diam ψ ( N t ) →
0. As ψ is a closed continuous embedding in theformer case, we can assume that we are in the latter.Let ψ be the extension of ψ to a partial function on N ≤ N ∗ given by ψ ( b ) = lim i →∞ ψ ( b ↾ i ) and ψ ( t a ( ∞ )) = lim i →∞ ψ ( t a ( i )) forall b ∈ N N and t ∈ N < N . By Proposition 5.2, we can assume that { ψ ( t a ( i )) | i ∈ N } has a limit point = ⇒ t a ( ∞ ) ∈ dom( ψ ) for all t ∈ N < N .As each point of N < N is isolated, diam ψ ( N b ↾ i ) → b ∈ N N ,and diam ψ ( N t a ( i ) ) → t ∈ N < N , it follows that ψ is continuous.To see that ψ is closed, it is sufficient show that every injective sequence( c n ) n ∈ N of points in the domain of ψ for which ( ψ ( c n )) n ∈ N is convergenthas a subsequence converging to a point in the domain of ψ . By passingto a subsequence, we can assume that the sequence converges to a pointof N ≤ N ∗ . As each point of N < N is isolated, the sequence converges toa point of N N ∗ , so the facts that diam ψ ( N b ↾ i ) → b ∈ N N ,diam ψ ( N t a ( i ) ) → t ∈ N < N , and { ψ ( t a ( i )) | i ∈ N } has alimit point = ⇒ t a ( ∞ ) ∈ dom( ψ ) for all t ∈ N < N ensure that itconverges to a point of the domain of ψ . HE FIRST BAIRE CLASS 15
By Proposition 2.8, we can assume that one of the following holds:(1) N N ∗ \ N N ⊆ dom( ψ ) and ∀ t ∈ N < N ψ ( t ) = ψ ( t a ( ∞ )).(2) N N ∗ \ N N ⊆ dom( ψ ) and ∀ t ∈ N < N ψ ( t ) = ψ ( t a ( ∞ )).(3) ( N N ∗ \ N N ) ∩ dom( ψ ) = ∅ .As the domain of ψ is analytic, so too is its intersection with N N . It fol-lows that the latter intersection has the Baire property, so Proposition2.9 allows us to assume that one of the following holds:(a) The domain of ψ is disjoint from N N .(b) The domain of ψ contains N N .In the special case that condition (b) holds, Theorem 3.6 allows us toassume that ψ ↾ N N is either constant or injective.Proposition 5.4 allows us to assume that ( ψ ( t )) t ∈ N < N converges tosome x ∈ X or for no natural numbers m < n is ψ ( t m ) or ψ ( t m a ( ∞ ))in the closure of ψ ( N t n ). In the former case, Proposition 5.1 allowsus to assume that ψ ( N < N ) is discrete, so the extension of ψ to N < N ∗ sending ∞ to x is a closed continuous embedding, thus we can assumethat we are in the latter. Lemma 5.6.
Suppose that c, d ∈ dom( ψ ) are distinct but ψ ( c ) = ψ ( d ) .Then there exists t ∈ N < N such that { c, d } = { t, t a ( ∞ ) } .Proof. To see that ψ ↾ N N ∗ \ N N is injective, observe that if m < n ,both t m a ( ∞ ) and t n a ( ∞ ) are in the domain of ψ , and moreover ψ ( t m a ( ∞ )) = ψ ( t n a ( ∞ )), then ψ ( t m a ( ∞ )) is in the closure of ψ ( N t n ), a contradiction.To see that ψ ↾ N N is injective when N N is contained in the domain of ψ , note that otherwise it is constant, and let x be this constant value.Then for each t ∈ N < N , there is a sequence ( u i ) i ∈ N of elements of N < N such that ψ ( t a ( i ) a ( u i )) → x , so the fact that diam ψ ( N t a ( i ) ) → ψ ( t a ( ∞ )) = x , contradicting the fact that ψ ↾ N N ∗ \ N N is injective.To see that ψ ( N N ) ∩ ψ ( N < N ) = ∅ , note that if b ∈ dom( ψ ) ∩ N N , t ∈ N < N , and ψ ( b ) = ψ ( t ), then there exist m < n with t m = t and t n ⊏ b , so ψ ( t m ) is in the closure of ψ ( N t n ), a contradiction.To see that ψ ( N N ) ∩ ψ ( N N ∗ \ N N ) = ∅ , note that if b ∈ dom( ψ ) ∩ N N , t ∈ N < N , t a ( ∞ ) ∈ dom( ψ ), and ψ ( b ) = ψ ( t a ( ∞ )), then there exist m < n with t m = t and t n ⊏ b , in which case ψ ( t m a ( ∞ )) is in theclosure of ψ ( N t n ), a contradiction.Observe finally that if s, t ∈ N < N are distinct, t a ( ∞ ) ∈ dom( ψ ),and ψ ( s ) = ψ ( t a ( ∞ )), then there exist m = n such that t m = s and t n = t . Then ψ ( t m ) is in the closure of ψ ( N t n ) and ψ ( t n a ( ∞ )) is in ψ ( N t m ), a contradiction.If (1a) or (1b) holds, then ψ ↾ N N ∗ \ N N or ψ ↾ N N ∗ is an extensionof φ to a closed continuous embedding. If (2a), (2b), (3a), or (3b)holds, then ψ is an extension of ψ to a closed continuous embeddingon N ≤ N ∗ \ N N , N ≤ N ∗ , N < N , or N ≤ N . Proposition 5.7.
Suppose that X is an analytic metric space, φ : N N ∗ \ N N → X , π : N < N → N < N is a ∧ -embedding, and φ ◦ π is constant, φ ◦ π extends to a closed continuous embedding on N N ∗ \ N N or N N ∗ ,or φ ◦ π ◦ p − extends to a closed continuous embedding on N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , or N ≤ N ∗ . Then there exist φ ∈ { c N N ∗ \ N N }∪{ ι N N ∗ \ N N ,Z | Z ∈{ N N ∗ \ N N , N N ∗ }}∪{ ι N < N ,Z ◦ p | Z ∈ { N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , N ≤ N ∗ }} and ψ : φ ( N N ∗ \ N N ) → φ ( N N ∗ \ N N ) with the property that ( π ↾ N N ∗ \ N N → N N ∗ \ N N , ψ ) is a closed continuous embedding of φ into φ .Proof. If φ ◦ π is constant, then set φ = c N N ∗ \ N N and let ψ be the uniquefunction from c N N ∗ \ N N ( N N ∗ \ N N ) to ( φ ◦ π )( N N ∗ \ N N ). If φ ◦ π extends toa closed continuous embedding ψ on Z ∈ { N N ∗ \ N N , N N ∗ } , then set φ = ι N N ∗ \ N N ,Z . And if φ ◦ π ◦ p − extends to a closed continuous embedding ψ on Z ∈ { N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , N ≤ N ∗ } , then set φ = ι N < N ,Z ◦ p .6. Borel functions that are not Baire class one
Here we provide bases for the classes of non-Baire-class-one Borelfunctions and non- σ -continuous-with-closed-witnesses Borel functionsbetween analytic metric spaces. Proposition 6.1.
Suppose that X is a metric space and φ : N N ∗ → X has the property that φ ↾ N N is continuous and φ ( N N ) * φ ( N N ∗ \ N N ) .Then there is a ∧ -embedding π : N < N → N < N with the property that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ .Proof. Fix b ∈ N N for which φ ( b ) is not in the closure of φ ( N N ∗ \ N N ).Then there is an open neighborhood U of φ ( b ) disjoint from φ ( N N ∗ \ N N ),as well as an open neighborhood V of φ ( b ) whose closure is containedin U , in which case the continuity of φ ↾ N N yields a proper initialsegment s of b for which φ ( N s ∩ N N ) ⊆ V . Then the ∧ -embedding π : N < N → N < N given by π ( t ) = s a t for all t ∈ N < N is as desired.Given φ N N : N N → X and φ N N ∗ \ N N : N N ∗ \ N N → Y , let φ N N ⊔ φ N N ∗ \ N N denote the corresponding function from N N ∗ to the disjoint union X ⊔ Y . HE FIRST BAIRE CLASS 17
Theorem 6.2.
Suppose that X and Y are analytic metric spaces and φ : X → Y is a Borel function that is not Baire class one. Then thereexist φ N N ∈ { c N N } ∪ { ι N N ,Z | Z ∈ { N N , N N ∗ }} and φ N N ∗ \ N N ∈ { c N N ∗ \ N N } ∪{ ι N N ∗ \ N N ,Z | Z ∈ { N N ∗ \ N N , N N ∗ }} ∪ { ι N < N ,Z ◦ p | Z ∈ { N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , N ≤ N ∗ }} for which there is a closed continuous embedding of φ N N ⊔ φ N N ∗ \ N N into φ .Proof. Hurewicz’s dichotomy theorem for F σ sets yields a closed contin-uous embedding ψ : N N ∗ → X with ( φ ◦ ψ )( N N ) ∩ ( φ ◦ ψ )( N N ∗ \ N N ) = ∅ .As ( ψ, id ( φ ◦ ψ )( N N ∗ ) ) is a closed continuous embedding of φ ◦ ψ into φ , byreplacing the latter with the former, we can assume that X = N N ∗ and φ ( N N ) ∩ φ ( N N ∗ \ N N ) = ∅ .By Proposition 3.1, there is a ∧ -embedding π : N < N → N < N for which( φ ◦ π ) ↾ N N is continuous. By composing π with the ∧ -embedding givenby Proposition 6.1, we can assume that ( φ ◦ π )( N N ) ∩ ( φ ◦ π )( N N ∗ \ N N ) = ∅ . By composing π with the ∧ -embedding given by Theorem 3.6, we canassume that ( φ ◦ π ) ↾ N N is constant or extends to a closed continuousembedding on N N or N N ∗ . And by composing π with the ∧ -embeddinggiven by Theorem 5.5, we can assume that ( φ ◦ π ) ↾ N N ∗ \ N N is constant,( φ ◦ π ) ↾ N N ∗ \ N N extends to a closed continuous embedding on N N ∗ \ N N or N N ∗ , or φ ◦ π ◦ p − extends to a closed continuous embedding on N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , or N ≤ N ∗ .By Proposition 3.7, there exist φ N N ∈ { c N N } ∪ { ι N N ,Z | Z ∈ { N N , N N ∗ }} and ψ N N : φ N N ( N N ) → φ ( N N ) for which ( π ↾ N N → N N , ψ N N ) is a closedcontinuous embedding of φ N N into φ ↾ N N . By Proposition 5.7, thereexist φ N N ∗ \ N N ∈ { c N N ∗ \ N N } ∪ { ι N N ∗ \ N N ,Z | Z ∈ { N N ∗ \ N N , N N ∗ }} ∪ { ι N < N ,Z ◦ p | Z ∈ { N < N , N < N ∗ , N ≤ N ∗ \ N N , N ≤ N , N ≤ N ∗ }} and ψ N N ∗ \ N N : φ N N ∗ \ N N ( N N ∗ \ N N ) → φ ( N N ∗ \ N N ) for which ( π ↾ N N ∗ \ N N → N N ∗ \ N N , ψ N N ∗ \ N N ) is a closedcontinuous embedding of φ N N ∗ \ N N into φ ↾ N N ∗ \ N N . Then ( π ↾ N N ∗ → N N ∗ , ψ N N ⊔ ψ N N ∗ \ N N ) is a closed continuous embedding of φ N N ⊔ φ N N ∗ \ N N into φ . Theorems 4.5 and 6.2 together provide the promised twenty-sevenelement basis under closed continuous embeddability for the class ofnon- σ -continuous-with-closed-witnesses Borel functions between ana-lytic metric spaces. References [CMS] R. Carroy, B.D. Miller, and D.T. Soukup,
The open dihypergraph dichotomyand the second level of the Borel hierarchy , to appear in ContemporaryMathematics. [JR82] J.E. Jayne and C.A. Rogers,
First level Borel functions and isomor-phisms , J. Math. Pures Appl. (9) (1982), no. 2, 177–205. MR 673304(84a:54072)[Kec95] A.S. Kechris, Classical descriptive set theory , Graduate Texts in Math-ematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597(96e:03057)[Sol98] S. Solecki,
Decomposing Borel sets and functions and the structure ofBaire class functions , J. Amer. Math. Soc. (1998), no. 3, 521–550.MR 1606843 (99h:26010) Rapha¨el Carroy, Kurt G¨odel Research Center for MathematicalLogic, Universit¨at Wien, W¨ahringer Straße 25, 1090 Wien, Austria
E-mail address : [email protected] URL : ∼ carroy/indexeng.html Benjamin D. Miller, Kurt G¨odel Research Center for Mathemati-cal Logic, Universit¨at Wien, W¨ahringer Straße 25, 1090 Wien, Aus-tria
E-mail address : [email protected] URL ::