Let be a topological space and an open cover thereof. A (continuous) path can pass through many of the elements of as it winds its way around . We can decompose that path into segments such that each segment lies wholly inside one of the open sets in . The Schedule Theorem says that this can be done continuously over all paths in .
This was proved by Dyer and Eilenberg and applied to the question of fibrations over numerable spaces.
The idea of a schedule is that it is a way of decomposing a length into pieces and then assigning a label to each piece. This clearly fits with the stated purpose of these things since we wish to decompose a path into pieces and assign an open set to each piece.
To make this precise, we start with a set of labels. Following Dyer and Eilenberg, let us write this as . Lengths are positive real numbers and so we also need the set of such, Dyer and Eilenberg denote this by ; thus .
A schedule in is thus a finite ordered list of pairs where and .
There are two notions of length for a schedule. There is the word length which simply counts the number of pairs. Then there is the function defined by . There is also a right action of on which simply multiplies all of the lengths: . Then .
The empty schedule is reduced.
There is a retraction map defined by removing all terms with zero length part.
The schedule monoid is given a topology so that the labels are discrete and the lengths topologised as usual. More concretely, given a word of elements in , the set of schedules of the form is in bijection with and we make that bijection a homeomorphism. Then is topologised by taking the coproduct over the set of words in . The reduced schedule monoid is topologised as the quotient of this.
Let be a topological space. Let denotes its Moore path space. Suppose that we have a family of subsets of indexed by some set . Then we consider a schedule in as giving an ordered list of these subsets together with the times to be spent in each. For a path in , and a schedule of the appropriate length, then we can ask whether or not the path fits (or obeys) the schedule. We make that precise as follows.
Here, is the function that assigns to a Moore path its length. The schedule designates a decomposition of into subintervals with being the length of the th subinterval. Then saying that fits the schedule means that spends the th subinterval in the open set .
We can now state the main theorem.
Let be a topological space. Let be a locally finite open covering of by numerable open sets with indexing set . Then there is a covering of by closed sets and a family of continuous functions , indexed by such that:
The first condition is purely about the covering. Dyer and Eilenberg use the term local covering for a covering by closed sets with this property.
There exists a continuous function such that if and if .
Here, is the empty word.
The original motivation for the notion of schedules was to prove the globalisation theorem for (Hurewicz) fibrations.
Let be a continuous function. Suppose that is a locally finite covering of by numerable open sets with the property that for each then the restriction is a fibration. Then is a fibration.
The link between the globalisation theorem and the schedule theorem is the characterisation of Hurewicz fibrations in terms of Hurewicz connections.
Let be a topological space. Let be a locally finite open covering of by numerable open sets and indexing set .
Let us write for the free monoid on . Then there is a function which takes to the schedule . We say that a path evenly fits , and write this as , if it fits the schedule corresponding to .
We need an initial technical result.
There is a locally finite covering of by numerable open sets such that for then evenly fits the word .
As is locally finite and its elements are numerable, we can choose a numeration that is also a partition of unity. That is, we can choose continuous functions with the property that and .
Let be the set of finite subsets of (where is the empty word). For we define
This is a covering of by closed sets. As is locally finite, for there is some neighbourhood which meets only a finite number of the . These are indexed by elements of , indeed of , and so the set of indices is an element, say , of. Then for , and so for , , whence . Thus each is contained in the interior of some .
Now let us put a total ordering on . This induces a total ordering on each and thus allows us to define the partial sums of the summation . Write these as , with as the zero function.
Fix and write it as in the inherited ordering. Let be a list of integers with the property that where is the word length of . Define:
This is closed in and the collection is a finite cover of . The family ranging over all and suitable is the family that we are looking for. It has the required covering property since the interiors of the cover .
Define as follows:
where is the schedule with and , and if then if we have
This is continuous and for then fits . Moreover, for then .
Dyer, E. and E., Samuel. (1988). Globalizing fibrations by schedules. Fund. Math., 130, 125–136. MR0963792
Dyer, Eilenberg, MR0963792