Smyths finite approximations University of Birmingham Chris Good, - - PowerPoint PPT Presentation

Smyth’s paper

Smyth’s finite approximations

University of Birmingham Chris Good, Mate Puljiz, David Parker, Jonathan E. Rowe Sheffield, April 4, 2014

Smyth’s paper Motivation Inverse limit construction The End

Universality

A space is universal for a certain class of spaces if all the spaces from that class can be realised as the quotients of the initial space. It is a well know fact that ωω is the universal Polish space (complete, metric, separable) and Cantor set is universal for the separable compact metric spaces. Eg, the binary expansion of a real number {0, 1}ω → [0, 1].

Smyth’s paper Motivation Inverse limit construction The End

Why this may be interesting to us?

Going back to the commuting diagram, U X U X Ξ ˜ f Ξ f starting with X we are provided with the map Ξ. If we could lift the map f to a map ˜ f on U then finding CGs of (X, f ) corresponds to finding CGs of (U, ˜ f ) that are constant on the level sets of Ξ.

Smyth’s paper Motivation Inverse limit construction The End

Problems?

Not every Ξ will work. It needs to be tuned with respect to the given f . Eg, the aforementioned binary expansion doesn’t allow lifting of the map x → 3

2x (hence, 1 3 → 1 2).

Can we always find Ξ that allows lifting?

Smyth’s paper Motivation Inverse limit construction The End

Construction

Inverse system X1

f2,1

← − X2

f3,2

← − X3

f4,3

← − X4 . . . Inverse limit lim

← − Xi = {(xi)i ∈

• i

Xi|fn+1,n(xn+1) ∈ xn}.

Smyth’s paper Motivation Inverse limit construction The End

Construction

Given a space X, take a sequence ǫn ↓ 0, construct finite (or countable) families of open sets Cn st. diameter of each set in Cn is less then ǫn sets from each Cn cover X each set from Cn+1 is contained in the unique set from Cn * image under f of each set in Cn+1 is contained in the unique set from Cn We think of the sets in Cn as symbols. The binding maps are

• inclusions. Assuming that X has no isolated points, the inverse

limit lim

← − Cn is homeomorphic to the Cantor set or ωω depending

• n whether X was compact or not. And f induces a lifted map ˜

f .

Smyth’s paper Motivation Inverse limit construction The End

A different perspective

Having constructed U = lim

← − Cn it is clear that the projection Ξ

should map (xi)i to xi where xi ∈ Ci is a decreasing sequence of

• sets. Also, the map is far from being 1-1 since sets in Ci are not

disjoint. Rather than constructing an universal covering space and then quotienting it to obtain initial space, Smyth encodes additional information within each finite approximation Ci in form of a directed graph with the vertices set Ci and with an edge connecting each pair of intersecting sets. This gives rise to a graph structure on the inverse limit. We put edge between (xi)i and (yi)i iff there is an edge connecting xi and yi for every i.

Smyth’s paper Motivation Inverse limit construction The End

A different perspective

Note that (xi)i and (yi)i are connected by an edge iff they represent the same point in X. Hence, space X is now given as an quotient with respect to this relation. The triplet (X, τ, R) is a topological graph which captures properties of both topological spaces and graphs. For a topological space one can take R to be the trivial (diagonal) relation, and for a finite graph, the topology is trivial (discrete) and the relation is

• ne given by the graph.

Idea is that on top of having a topology, the relation R tells us which are our the closest neighbours of a given point. Using this

• ne can construct a coarser topology on X consisting of only those
• pen sets such that they have their close neighbours inside that set

(R-invariant). The intermediate step is taking all such (not just

• pen) sets and that family needs not to form a topology but it is a

neighbourhood space.

Smyth’s paper Motivation Inverse limit construction The End

This essentially means that graphs are represented as non Hausdorff finite spaces and their inverse limit is a non Hasdorff space whose T0 quotient is initial space. ǫ - approximation by a finite graph then amounts to taking Cn consisting of subsets having small enough diameters. The trouble is that our lifted function is not mapping on Cn. It is mapping from Cn+1 to Cn which means that mapping is increasing

• ur error. After n iterations we will not know where in the space

we are.

Smyth’s paper Motivation Inverse limit construction The End

The End!

Thank you!