Overt Subspaces of Locally Compact Metric Spaces Paul Taylor PSSL - - PowerPoint PPT Presentation

Overt Subspaces

• f Locally Compact Metric Spaces

Paul Taylor PSSL 94 (BMC) University of Sheffield Monday 25 March 2013 Introducing overt subspaces to the general mathematician and characterising them in Rn. 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 difference 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

• ften 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 Hausdorff target. Ho hum.

Here is a simple classical theorem

A continuous function f : X → Y between locally compact metric spaces is an open map iff the expression dy(x) ≡ inf {d(x, a) | f(a) = y} defines a continuous function d(−)(−) : X × Y → R. In this case, f(x) = y ⇐⇒ dy(x) = 0. This ought to be well known. Is it? Without local compactness, f is open iff dy(x) is upper semicontinuous in y.

Illustrative non-example

For the squaring map f : R → R, f(x) = x2 and dy(x) =

• |x| − √y
• if y ≥ 0

∞ if y < 0, but dy(x) is not continuous in y near 0.

Locatedness

The same idea using a subset instead of a parameter, dA(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 −1V 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: ♦

• i

Ui ⇐⇒ ∃i. ♦ Ui. 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 ♦ : ΣX → Σ. where ΣX 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 dA(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 dA(x) and so in the relation dA(x) < r.

Locatedness again

Recall that a subspace A is located if dA(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 dA(x) and so in the relation dA(x) < r. This is rounded, convergent and satisfies the triangle law: d(x) < r ⇐⇒ ∃r′. d(x) < r′ < 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 ≡ ♦ Br(x). This too is rounded, convergent and satisfies the triangle law: d(x) < r ⇐⇒ ∃r′. d(x) < r′ < 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 ∧ Br(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)

• vert subspaces of locally compact metric spaces in terms of

upper semicontinuous metric-like functions.

Accumulation points

By roundedness and convergence, if d(xn) < rn then d(xn, xn+1) < rn − rn+1 ∧ d(xn+1) < rn+1 < 1

2rn

for some xn+1 and rn+1. Thus there are sequences x0, x1, . . . and r0, r1, . . . with Br0(x0) ⊃ · · · ⊃ Brn(xn) ⊃ Brn+1(xn+1) ⊃ Brn+1(xn+1) ⊃ · · · d(xn, xn+k), d(xn) < rn < 2−nr0. Hence (xn) is a Cauchy sequence, with limit a ≡ x∞ and d(xn, a) < rn. So d(a) < 2rn < 2−n+1r0 for all n, whilst d(x0, a) < r0. 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,

• r

∃xr. d(x) < r ∧ Br(x) ⊂ U, then U contains an accumulation point a of ♦ or d. That is, a ∈ U with d(a) = 0

• r

∀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 Gδ (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 iff 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 | dy(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 dy(x) is jointly continuous X × Y → R. If this holds then X {(x, y) | dy(x) = 0}

π1

✲ Y

is an open map.

The Newton–Raphson algorithm

Let f : Rn → Rn be continuously differentiable with ˙ f(x) invertible. Then the sequence xn+1 ≡ xn + g(xn) where g(x) ≡ ˙ f(x) −1 ·

• y − f(x)
• .

hopefully converges to a ≡ x∞ with f(a) = y. The function

• g(x)
• has properties similar to our dy(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.

Kantorovich convergence of Newton–Raphson

This algorithm can behave chaotically. In our setting,

• g(x)
• need not obey the convergence property.

We rectify this by incorporating the convergence criteria of the Newton–Raphson algorithm (Kantorovich, 1948) into the logical definition.

Kantorovich convergence of Newton–Raphson

This algorithm can behave chaotically. In our setting,

• g(x)
• need not obey the convergence property.

We rectify this by incorporating the convergence criteria of the Newton–Raphson algorithm (Kantorovich, 1948) into the logical definition. We define ∆(x) < r to hold if

◮ ˙

f(x) is invertible,

◮

• ˙

f(x)−1 · (y − f(x))

• < 1

2r and ◮ ∀x′x′′ ∈ Br(x).

• ˙

f(x)−1 · ˙ f(x′) − ˙ f(x′′)

• ≤ |x′ − x′′|/r.

Otherwise we put ∆(x) ≡ ∞. Adapting a recent proof by Philippe Ciarlet and Cristinel Mardare, ∆(x) < r =⇒ ∆(x1) < 1

2r

where x1 ≡ x − ˙ f(x)−1 · f(x).

Algorithms like Newton–Raphson

When we define ∆(x) from the Newton-Raphson algorithm and its convergence criterion, it satisfies roundedness and convergence but not the triangle law. So we define d(x) < r as ∃ys. ∆(y) < s ∧ d(x, y) < r − s This then satisfies all three axioms for d. In fact, we can often just use ∆ instead of d.

Numerical algorithms in general

With some massaging, an algorithm that estimates how far away the nearest solution is provides an example of our d function. Conversely, a d function with a realiser for its convergence property provides an algorithm for finding accumulation points. Solvable problems are encoded as d or ♦. Solutions are accumulation points. The Tangency Theorem guarantees their existence. The extent (set of solutions) plays essentially no role in this. An overt subspace is defined by ♦ or d not A.

Nets

A net is a continuous function f : M → X where M is a computationally representable set. The precise definition depends on your particular foundational setting. But the idea is very familiar. Any numerical computation is a net, where M is the type of (necessarily discrete) representations of the input data for the problem and initial approximation. Any sequence is a net. Countable dense subspaces are common in analysis.

Nets and overt subspaces

Any net defines an overt subspace by MU ≡ ∃m ∈ M. f(m) ∈ U. dM(x) < r ≡ ∃m ∈ M. d(x, f(m)) < r.

Nets and overt subspaces

Any net defines an overt subspace by MU ≡ ∃m ∈ M. f(m) ∈ U. dM(x) < r ≡ ∃m ∈ M. d(x, f(m)) < r. The Tangency Theorem gives the converse: Let M be the set of rational coordinates (x0, r0)

• f balls Br0(x0) such that d(x0) < r0 ≡ ♦ Br0(x0) holds.

Let f(x0, r0) ∈ X be the accumulation point a that is the limit of the sequence of approximants starting at (x0, r0). Then f : M → X is a net that yields the given ♦ or d. Beware that f is not determined uniquely.

Characterisation of overt subspaces

We have proved the equivalence amongst

◮ a join-preserving operator ♦, ◮ an upper distance function d and‘ ◮ a net f : M → X.

We expect the direct image of any overt subspace to be overt,

• cf. compact ones.

Therefore our working definition, that ♦ just be a join-preserving operator, is sufficient: any such operator defines an overt subspace. Beware that this Theorem has different logical strengths for classical point-set topology and locale theory, intuitionistic locale theory, formal topology and abstract Stone duality.

Existential quantifier

Other ways of stating the Tangency Theorem are ♦ U ⇐⇒ ∃a ∈ A. a ∈ U d(x) < r ⇐⇒ ∃a ∈ A. d(x, a) < r. An overt subspace is one for which the existential quantifier is meaningful. The Tangency Theorem is a special case of Gentzen’s Existence Theorem, for the logic (ASD) of open subspaces.

Demarcation between mathematics and computation

To a pure mathematician,

• vertness is a topological property like compactness.

The Tangency Theorem says that if ♦ U holds then some accumulation point exists in A ∩ U. Where “exists” has its usual aetherial meaning.

Demarcation between mathematics and computation

To a pure mathematician,

• vertness is a topological property like compactness.

The Tangency Theorem says that if ♦ U holds then some accumulation point exists in A ∩ U. Where “exists” has its usual aetherial meaning. To a programmer ♦ U is a specification of a computation, to find such a point. Such a computation may be either numerical analysis or logic programming.