Overt Subspaces of Locally Compact Metric Spaces Paul Taylor PSSL 94 (BMC) University of She ffi eld Monday 25 March 2013 Introducing overt subspaces to the general mathematician and characterising them in R n . In memory of my late parents and funded by their savings.
Overtness has arisen in many ways Open maps between intuitionistic locales or in formal topology. Positivity in intuitionistic locales or in formal topology. Locatedness in constructive analysis. Existential quantifiers. Recursive enumerability. Dense nets in analysis. Algorithms such as Newton–Raphson. Evidence for solutions of problems. But it is invisible in classical point–set topology. (Or even classical locale theory.)
Just tell me the f—ing definition! I only wish I could. Excluded middle is just a tiny part of the problem. It would be a start if “ordinary” mathematicians would acknowledge the di ff erence amongst lower, upper and Euclidean reals. Overtness is about having evidence that a subspace has some points. So point–set topology gets in the way. In fact this treatment was inspired by that of Bas Spitters, whose setting was Formal Topology. Overtness is also about which unions and existential quantifiers are valid. So we cannot rigorously use ordinary mathematical language to define and study overtness without already having defined and understood overtness.
The ghost of constructivity in classical topology The notion of overtness is essentially constructive. How can we introduce it using classical point–set topology? Classical mathematics tries hard to smother constructive issues. However, they re-emerge in other guises. Introducing continuous parameters often involves the same reasoning as constructivity. An overt subspace is a fibre of an open map. But any closed subspace can be a fibre of an overt map with Hausdor ff target. Ho hum.
Here is a simple classical theorem A continuous function f : X → Y between locally compact metric spaces is an open map i ff the expression d y ( x ) ≡ inf { d ( x , a ) | f ( a ) = y } defines a continuous function d ( − ) ( − ) : X × Y → R . In this case, f ( x ) = y ⇐⇒ d y ( x ) = 0 . This ought to be well known. Is it? Without local compactness, f is open i ff d y ( x ) is upper semicontinuous in y .
Illustrative non-example For the squaring map f : R → R , � | x | − √ y � � � if y ≥ 0 � � f ( x ) = x 2 and d y ( x ) = � ∞ if y < 0 , but d y ( x ) is not continuous in y near 0.
Locatedness The same idea using a subset instead of a parameter, d A ( x ) ≡ inf { d ( x , a ) | a ∈ A } , is used very frequently in constructive analysis. The subspace A is called located if this inf is defined as a (Euclidean) real number, not just as a lower real number. Often, locatedness has to be added as a hypothesis to make standard theorems valid constructively. For example, the kernel of a bounded linear map is located, so this is needed to construct quotients. We need to get ordinary mathematicians to understand the distinction between Euclidean reals and upper reals.
Parameters to the rescue again A function d is upper semicontinuous if { x | d ( x ) < r } ⊂ X is open for each r ∈ Q . This is recognised as a mainstream definition, albeit often in a complicated form using lim sup. The values of an upper semicontinuous function are upper reals. An upper real is an upper semicontinuous constant value.
Open maps between locales (frames) f : X → Y is open if there is an operator f ! on the frames such that f ! U ⊂ V ⇐⇒ U ⊂ f − 1 V plus two other conditions: ◮ Frobenius: f ! commutes with intersections; ◮ Beck–Chevalley: f ! commutes with inverse images. How do you state Beck–Chevalley without introducing pullbacks? However, we can avoid talking about Frobenius and Beck–Chevalley in our context.
Overt subspaces as ♦ For an open map f : X → Y , a point y ∈ Y and an open subspace U ⊂ X , U touches (has non-trivial intersection with) the fibre f − 1 ( y ) if ♦ U ≡ y ∈ f ! U . Then ♦ preserves joins: � U i ⇐⇒ ∃ i . ♦ U i . ♦ i We take such an operator ♦ that preserves joins as our working definition of an overt subspace.
Overt subspaces as ♦ We take such an operator ♦ that preserves joins as our working definition of an overt subspace. It is a working definition because we have ignored Frobenius and Beck–Chevalley and also whether it should be a continuous function ♦ : Σ X → Σ . where Σ X is an exponential in some category of spaces. In the setting of locally compact metric spaces, every join-preserving operator arises from some situation that we would agree justifies the name overt. That is deliberately vague and will become clear later.
Locatedness again Recall that a subspace A is located if d A ( x ) ≡ inf { d ( x , a ) | a ∈ A } , is defined as a (Euclidean) real number, for each x ∈ X . The constructive content lies in the upper value of d A ( x ) and so in the relation d A ( x ) < r .
Locatedness again Recall that a subspace A is located if d A ( x ) ≡ inf { d ( x , a ) | a ∈ A } , is defined as a (Euclidean) real number, for each x ∈ X . The constructive content lies in the upper value of d A ( x ) and so in the relation d A ( x ) < r . This is rounded, convergent and satisfies the triangle law: ∃ r ′ . d ( x ) < r ′ < r d ( x ) < r ⇐⇒ = ⇒ ∃ y . d ( y ) < ǫ ∧ d ( x , y ) < r = ⇒ d ( x ) < r + ǫ for any ǫ > 0. If a ∈ A then d ( a ) = 0. If A is also closed then d ( x ) = 0 = ⇒ x ∈ A .
Overt subspaces of metric spaces The technical content of the talk starts here. Now start from any join-preserving operator ♦ . Define d ( x ) < r ♦ B r ( x ) . ≡ This too is rounded, convergent and satisfies the triangle law: ∃ r ′ . d ( x ) < r ′ < r d ( x ) < r ⇐⇒ = ⇒ ∃ y . d ( y ) < ǫ ∧ d ( x , y ) < r = ⇒ d ( x ) < r + ǫ These properties make { x | d ( x ) < r } open. Then d : X → R is upper semicontinuous and d ( x ) ≡ inf { r | d ( x ) < r } is an upper real number.
Correspondence between d and ♦ In a locally compact metric space, we recover ♦ U ⇐⇒ ∃ xr . d ( x ) < r ∧ B r ( x ) ⊂ U . Alternatively, if we define ♦ from d like this, it preserves joins. Moreover the correspondence is bijective between ◮ ♦ preserving joins and ◮ d : X → R rounded, convergent and with the triangle law. Hence we have a characterisation of (not necessarily closed) overt subspaces of locally compact metric spaces in terms of upper semicontinuous metric-like functions.
Accumulation points By roundedness and convergence, if d ( x n ) < r n then d ( x n + 1 ) < r n + 1 < 1 d ( x n , x n + 1 ) < r n − r n + 1 ∧ 2 r n for some x n + 1 and r n + 1 . Thus there are sequences x 0 , x 1 , . . . and r 0 , r 1 , . . . with B r 0 ( x 0 ) ⊃ · · · ⊃ B r n ( x n ) ⊃ B r n + 1 ( x n + 1 ) ⊃ B r n + 1 ( x n + 1 ) ⊃ · · · < r n < 2 − n r 0 . d ( x n , x n + k ) , d ( x n ) Hence ( x n ) is a Cauchy sequence, with limit a ≡ x ∞ and d ( x n , a ) < r n . So d ( a ) < 2 r n < 2 − n + 1 r 0 for all n , whilst d ( x 0 , a ) < r 0 . We define d ( a ) = 0 to mean this and call a an accumulation point of d .
Tangency Theorem If an overt subspace defined by ♦ or d touches an open subspace U , ♦ U , or ∃ xr . d ( x ) < r ∧ B r ( x ) ⊂ U , then U contains an accumulation point a of ♦ or d . That is, a ∈ U with d ( a ) = 0 or ∀ V . a ∈ V = ⇒ ♦ V .
The subspace of accumulation points The extent of ♦ or d is the subspace of accumulation points. It is sequentially closed. It is G δ (countable intersection of opens). It need not be topologically closed, the complement of an open subspace. I do not understand the definition of weakly closed but if it means what I think it does then the extent is weakly closed too.
Closed overt subspaces In our characterisation, ◮ d ( x ) need only be an upper real (not Euclidean) and ◮ the extent A need not be closed. d : X → R has Euclidean values i ff the extent A is closed. Previous similar work ( e.g. Bas Spitters) has assumed these. Overt compact subspaces are particularly well behaved.
Open maps What if d has a continuous parameter y : Y ? We just require upper semicontinuity, so { y | d y ( x ) < r } is to be open. If we define ♦ and d from an open map then this is satisfied. If X and Y are locally compact metric spaces then d y ( x ) is jointly continuous X × Y → R . If this holds then π 1 ✲ Y X � { ( x , y ) | d y ( x ) = 0 } is an open map.
The Newton–Raphson algorithm Let f : R n → R n be continuously di ff erentiable with ˙ f ( x ) invertible. Then the sequence � − 1 · � ˙ � � x n + 1 ≡ x n + g ( x n ) where g ( x ) ≡ f ( x ) y − f ( x ) . hopefully converges to a ≡ x ∞ with f ( a ) = y . � � � has properties similar to our d y ( x ): The function � g ( x ) � � ◮ it is rounded (upper semicontinuous) in x and y , because it is actually continuous; and ◮ it has the convergence property in favourable circumstances; but ◮ it usually does not satisfy the triangle law.
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