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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 .
The schedule monoid of is the free monoid on the set . It is written . Its elements are schedules in .
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 .
A schedule is said to be reduced if all of its terms, , have non-zero length, i.e. . The set of reduced schedules forms a submonoid of which is written .
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.
Suppose that we have and , and suppose that . Then we say that fits the schedule s, written , if the following conditions hold:
- We can split into subpaths according to the times . Let be the th segment. Then .
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:
- for each , there some finite subfamily such that is in the interior of ,
- for each , , and
- for each ,
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.
Proof of the Schedule Theorem
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