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 on 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 ω 1 .

Some examples ◮ χ Z is Baire class 1. ⋆ ◮ χ Q is Baire class 2 and not Baire class 1. ◮ The derivative of any differential function on R is Baire class 1. f ( x + 1 /n ) − f ( x ) → f ′ ( x ) • f n ( x ) ≡ − 1 /n Consider f ( x ) = x 2 sin ( 1 /x ) , with f ( 0 ) ≡ 0. ◮ Theorem — Lebesgue For every countable ordinal β there is a real-valued function on 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 one 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 X d . ◮ But all functions are continuous on X d . Can the Baire functions on X be regarded as the continuous functions on 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 o 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 D Y which separates the points of Y . ◮ D Y 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 D Y 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 on 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 ⊆ R L which is cozero dense , i.e., � � � coz � g : g ∈ G, coz � g = � g ( R � {0} ) ≤ x for all x ∈ L . L is x = unique up to isomorphism with respect to this property. ◮ R L is the family of frame maps O R → L . It is a W -object.

The localic representation of W -objects has certain advantages ◮ We can avoid the difficulties posed by functions taking on 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 ⊆ R L which is cozero dense , i.e., � � � coz � g : g ∈ G, coz � g = � g ( R � {0} ) ≤ x for all x ∈ L . L is x = unique up to isomorphism with respect to this property. ◮ R L is the family of frame maps O R → L . It is a W -object.

The localic representation of W -objects has certain advantages ◮ We can avoid the difficulties posed by functions taking on 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 ⊆ R L which is cozero dense , i.e., � � � coz � g : g ∈ G, coz � g = � g ( R � {0} ) ≤ x for all x ∈ L . L is x = unique up to isomorphism with respect to this property. ◮ R L is the family of frame maps O R → L . It is a W -object.

The localic representation of W -objects has certain advantages ◮ We can avoid the difficulties posed by functions taking on 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 ⊆ R L which is cozero dense , i.e., � � � coz � g : g ∈ G, coz � g = � g ( R � {0} ) ≤ x for all x ∈ L . L is x = unique up to isomorphism with respect to this property. ◮ R L is the family of frame maps O R → 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 f 1 ≥ f 2 ≥ f 3 . . . of continuous nonnegative real-valued functions on X .

� f n = 0 means ¬∃ f 0 ∀ n ( 0 < f 0 ≤ f n ) R

� • f n = 0 means � n f − 1 n ( − ∞ , ε ) = X for all ε > 0 R

The definition of pointwise downwards convergence in W Definition ◮ A sequence { g n } ⊆ R L + converges pointwise downwards � • to 0, written g n ց 0, if it is decreasing and n g n = 0, i.e., � n g n ( − ∞ , ε ) = ⊤ for all ε > 0. ◮ Dually, a sequence { g n } ⊆ R L converges pointwise upwards to 0, written g n ր 0, if it is increasing and � n g n ( − ε, ∞ ) = ⊤ for all ε > 0. ◮ For an arbitrary decreasing sequence { g n } and element g 0 of R L , g n ց g 0 if ( g n − g 0 ) ց 0. Similarly, g n ր g 0 is defined dually. ◮ For a sequence { g n } and element g 0 in a W -object G , we say that g n converges pointwise downwards to g 0 , and write g n ց g 0 , if � g n ց � g 0 in R L , where g → � g ∈ R L is the Madden representation of G .