Pointfree pointwise convergence, Baire functions, and epimorphisms - - PowerPoint PPT Presentation

Pointfree pointwise convergence, Baire functions, and epimorphisms in truncated archimedean ℓ-groups

• R. N. Ball

University of Denver

10 August 2018

The classical Baire functions

◮ Definition — René-Louis Baire, 1899

The Baire functions comprise the least class of real-valued functions on a T ychonoff space X which

◮ contains the continuous functions, and ◮ is closed under pointwise convergence.

◮ Inductive Definition

◮ The functions of Baire class 0 are the continuous functions

• n X.

◮ The functions of Baire class β are the pointwise limits of

sequences of functions of Baire class α < β.

◮ The Baire functions on X are the functions of Baire class α

for some α. Alternatively, the functions of Baire class ω1.

Some examples

◮ χZ is Baire class 1.⋆ ◮ χQ is Baire class 2 and not Baire class 1. ◮ The derivative of any differential function on R is Baire

class 1. fn(x) ≡ f(x + 1/n) − f(x) 1/n

• −

→ f ′(x) Consider f(x) = x2 sin(1/x), with f(0) ≡ 0.

◮ Theorem — Lebesgue

For every countable ordinal β there is a real-valued function

• n R of Baire class β but not Baire class α for any α < β.

◮ Baire Characterization Theorem

A real valued function on the unit interval is Baire class 1 iff its restriction to any nonempty closed subset has at least

• ne point of continuity (relative to the subset).

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

Where do the Baire functions live?

◮ Can the Baire functions on a space X be regarded as

continuous on some other space associated with X?

◮ Yes, on the discrete space Xd. ◮ But all functions are continuous on Xd. Can the Baire

functions on X be regarded as the continuous functions

• n some space associated with X?

◮ Is there a space on which the Baire functions are

continuous and generate the topology of the space?

◮ No

How do we get a handle on the Baire functions?

◮ The Baire functions on a T

ychonoff space X form an archimedean ℓ-group which includes the constant

• functions. That is to say the Baire functions constitute a

good example of a truncated archimedean ℓ-group.

◮ For the purposes of this talk, we will confine our

attention to the classical category W of (weak) unital archimedean ℓ-groups.

◮ T

• represent W-objects, we have the classical

Hager-Robertson representation theorem.

Theorem — Hager, Robertson, 1977

For every W-object G there is a unique compact Hausdorff space Y such that G is isomorphic to a W-object in DY which separates the points of Y.

◮ DY is the family of continuous extended-real valued

functions on Y which are almost finite, i.e., finite on a dense open subset of Y.

◮ Note that DY is not always itself a W-object. ⋆

The localic representation of W-objects has certain advantages

◮ We can avoid the difficulties posed by functions taking

• n the values ±∞ by simply removing the points where

the infinities occur from the representation space.

◮ Of course, there may be no points left. ◮ That’s OK. Every locale has a minimum dense sublocale. ◮ This leads directly to the Madden representation for

W-objects.

◮ Theorem — Madden, Vermeer, 1990

For every W-object G there is a unique regular Lindelöf locale L such that G is isomorphic to a subobject G ⊆ RL which is cozero dense, i.e., x = coz g : g ∈ G, coz g = g(R {0}) ≤ x

• for all x ∈ L. L is

unique up to isomorphism with respect to this property.

◮ RL is the family of frame maps OR → L. It is a W-object.

The localic representation of W-objects has certain advantages

◮ We can avoid the difficulties posed by functions taking

• n the values ±∞ by simply removing the points where

the infinities occur from the representation space.

◮ Of course, there may be no points left. ◮ That’s OK. Every locale has a minimum dense sublocale. ◮ This leads directly to the Madden representation for

W-objects.

◮ Theorem — Madden, Vermeer, 1990

For every W-object G there is a unique regular Lindelöf locale L such that G is isomorphic to a subobject G ⊆ RL which is cozero dense, i.e., x = coz g : g ∈ G, coz g = g(R {0}) ≤ x

• for all x ∈ L. L is

unique up to isomorphism with respect to this property.

◮ RL is the family of frame maps OR → L. It is a W-object.

The localic representation of W-objects has certain advantages

◮ We can avoid the difficulties posed by functions taking

• n the values ±∞ by simply removing the points where

the infinities occur from the representation space.

◮ Of course, there may be no points left. ◮ That’s OK. Every locale has a minimum dense sublocale. ◮ This leads directly to the Madden representation for

W-objects.

◮ Theorem — Madden, Vermeer, 1990

For every W-object G there is a unique regular Lindelöf locale L such that G is isomorphic to a subobject G ⊆ RL which is cozero dense, i.e., x = coz g : g ∈ G, coz g = g(R {0}) ≤ x

• for all x ∈ L. L is

unique up to isomorphism with respect to this property.

◮ RL is the family of frame maps OR → L. It is a W-object.

The localic representation of W-objects has certain advantages

◮ We can avoid the difficulties posed by functions taking

• n the values ±∞ by simply removing the points where

the infinities occur from the representation space.

◮ Of course, there may be no points left. ◮ That’s OK. Every locale has a minimum dense sublocale. ◮ This leads directly to the Madden representation for

W-objects.

◮ Theorem — Madden, Vermeer, 1990

For every W-object G there is a unique regular Lindelöf locale L such that G is isomorphic to a subobject G ⊆ RL which is cozero dense, i.e., x = coz g : g ∈ G, coz g = g(R {0}) ≤ x

• for all x ∈ L. L is

unique up to isomorphism with respect to this property.

◮ RL is the family of frame maps OR → L. It is a W-object.

Every W-object has a home Question: where do the Baire functions live?

How do we talk about pointwise convergence in a pointfree setting?

◮ In order to get a handle on pointwise convergence, we

need to express pointwise infima of real-valued functions in a pointfree setting.

◮ Consider a descending sequence f1 ≥ f2 ≥ f3 . . . of

continuous nonnegative real-valued functions on X.

• fn = 0 means ¬∃f0 ∀n (0 < f0 ≤ fn)

R

• fn = 0 means
• n f −1

n (−∞, ε) = X for all ε > 0 R

The definition of pointwise downwards convergence in W

Definition

◮ A sequence {gn} ⊆ RL+ converges pointwise downwards

to 0, written gn ց 0, if it is decreasing and

• n gn = 0, i.e.,
• n gn(−∞, ε) = ⊤ for all ε > 0.

◮ Dually, a sequence {gn} ⊆ RL converges pointwise

upwards to 0, written gn ր 0, if it is increasing and

• n gn(−ε, ∞) = ⊤ for all ε > 0.

◮ For an arbitrary decreasing sequence {gn} and element

g0 of RL, gn ց g0 if (gn − g0) ց 0. Similarly, gn ր g0 is defined dually.

◮ For a sequence {gn} and element g0 in a W-object G, we

say that gn converges pointwise downwards to g0, and write gn ց g0, if gn ց g0 in RL, where g → g ∈ RL is the Madden representation of G.

The pointwise infima are context free

Theorem

Let {gn} be a descending sequence in G+ for some W-object

• G. Then gn ց 0 iff
• n θ(g) = 0 for every W-morphism

θ: G → H.

Directional pointwise convergence interacts nicely with W

◮ Theorem

The operations of W are directionally pointwise continuous.

◮ ˙

g ց g, where { ˙ g} is the sequence whose every term is g.

◮ gn ց g0 ⇐⇒ (−gn) ր (−g0). ◮ If gn ց g0 and fn ց f0 then (gn ⊕ fn) ց (g0 ⊕ f0), where ⊕

stands for +, ∨, or ∧.

◮ If gn ց g0 and gn ց f0 then f0 = g0.

◮ Theorem

Every W-homomorphism θ: G → H is directionally pointwise

• continuous. That is, gn ց g0 in G implies θ(gn) ց θ(g0) in H.

◮ Corollary

Let G ≤ H in W. If G is directionally pointwise dense in H then G is epically embedded in H. This is on the surface.⋆

A big question.

◮ Is G directionally pointwise dense in every epic

extension? We conjecture that it is.

◮ Let us ask a smaller question. If L is the Madden frame of

G then G ≡ G is obviously epically embedded in RL. Is G directionally pointwise dense in RL?

◮ Lemma

Every element of R+L is the pointwise join of a countable subset of G+. ⋆

◮ The proof is based on the fact that

G is cozero dense in L, i.e., x = coz g : g ∈ G, coz g ≥ x

• for all x ∈ L. From the

lemma it is easy to prove the following.

◮ Theorem

G is directionally pointwise dense in RL.

Epimorphisms in W

Theorem

W admits a functorial epicompletion. That is, every G ∈ W has an epic extension G → E such that E has no proper epic extensions, and this extension is functorial, i.e., a reflection.

◮ Theorem

The epicomplete objects in W are those of the form RP for P a P-frame.

◮ Definition

A P-frame is a frame in which each cozero element is complemented.

◮ Theorem

P-frames form a full monoreflective subcategory of the category of frames, hence we have a P-frame reflection L → PL for each frame L.

◮ Theorem

The extension G = G → RL → RPL is the functorial epicompletion of G in W.

More about the epicompletion G → RPL of G

◮ The P-frame reflection L → PL can be understood as the

colimit of a transfinite sequence L = L0 → L1 → L2 . . . PL, where Ln → Ln+1 results from freely complementing the cozero elements of cozLn, a standard construction in pointfree topology.⋆

◮ This immediately gives a parallel sequence of extensions

• f G.

G → RL → RL1 → RL2 . . . RPL Notation: G → G0 → G1 → G2 → . . . Gω1

One step

Lemma

Let M → N be a frame extension such that the cozero element u ∈ M is complemented in N. Then the characteristic function χu of u is an element of RN, and there is a sequence {gn} ⊆ RM such that gn ր χu. ⋆

Lemma

◮ For each sequence {gn} ⊆ Gn which is increasing and

bounded above in Gn+1 there is a unique element h ∈ Gn+1 such that gn ր h.

◮ Each element of Gn+1 is the pointwise join of pointwise

meets of elements of Gn.

Theorem

G is directionally pointwise dense in Gω1, i.e., in its

• epicompletion. Consequently G is pointwise dense in any

epicompletion.

The Big Question again: is every epic extension a directionally pointwise dense extension?

◮ Theorem

Any extension G → H in which G is directionally pointwise dense is an epic extension. And for any epic extension G → H there is an epic extension H → K such that G is directionally pointwise dense in K. ⋆

Pointwise convergence, the lim sup definition

We define unqualified pointwise convergence of a sequence by means of the familiar lim sup definition, i.e., by requiring the downward pointwise convergence of the suprema of the tails of the sequence.

Definition

A sequence {gn} ⊆ G+, bounded in G1, converges pointwise to 0, written gn

• −

→ 0, provided that

• j≥n

gj ց 0 in G1. For an arbitrary sequence {gn} and element g0 of G, gn

• −

→ g0 if |gn − g0| • − → 0 in G1. (Recall that bounded sequences in G+ have pointwise limits in G1.)

Pointwise convergence interacts nicely with W

Theorem

The operations of W are pointwise continuous.

Theorem

Every W-homomorphism is pointwise continuous.

Pointwise Cauchy sequences

◮ Definition

A sequence {gn} ⊆ G is said to be pointwise Cauchy if it is bounded in G1 and

• i,j≥m

(gi − gj) ց 0, in G2.

◮ Proposition

A sequence {gn} ⊆ G, bounded in G1, is pointwise Cauchy iff

• m
• n≥m

gn =

• m
• n≥m

gn in G2. .

The Baire functions on a locale

◮ Definition

For any W-object H, G ≤ H ≤ RPL, let H• ≡

• k ∈ RPL : ∃{hn} ⊆ H (hn
• −

→ k)

• .

◮ Definition

For a frame L, define the Baire functions on L as follows. B0L ≡ RL, BβL ≡

α<β

BαL

• BL ≡ Bω1L

The two sequences are intertwined

Lemma

• 1. B1L ⊆ G2 and G1 ⊆ B2L.
• 2. Bβ+nL ⊆ Gβ+2n and Gβ+n ⊆ Bβ+2nL for ordinals of the form

β + n, β a limit and n finite.

• 3. BβL = Gβ for limit ordinals β.
• 4. BL = Gω1 = RPL.

Characterizations of epicomplete W-objects

Theorem

The following are equivalent for a W-object G.

• 1. G is pointwise inextensible, i.e., G has no proper

extension in which it is pointwise dense.

• 2. G is pointwise Cauchy complete, i.e., every pointwise

Cauchy sequence is convergent.

• 3. G is epicomplete, i.e., G has no proper epic extension.
• 4. G is of the form RP for a P-frame P.
• 5. G is divisible and both laterally and conditionally

σ-complete.

• 6. G is DX for some compact basically disconnected

Hausdorff space X.

• 7. G can be uniquely endowed with a multiplication making

it an archimedean f-ring with identity which is uniformly complete and regular.

Summary

Theorem

For any frame L, the Baire functions on L may be identified with the continuous functions on PL.

A remark on P-spaces

Definition

A space X is a P-space if each continuous real-valued function on X is locally constant, i.e., if CX = LCX.

Proposition

X is a P-space iff each cozero subset of X is clopen.

Theorem

P spaces from a monocoreflective subcategory of completely regular spaces. But points of the P-frame reflection of a locale L are in bijective correspondence with those of L. Therefore, if L is spatial but not a P-frame, its P-frame reflection is not spatial. So we have answered the question of where the Baire functions live, but they live on a locale which is seldom a space.

Thank you!