Today, for the first Theory Lunch after the summer holidays, I happened to talk about the subject of some notes I am taking for myself.
Let us start by clarifying the context. Recall that a topology on a set is a family of subsets of such that:
- if then ; and
- if then .
For instance, the family made of all the unions of open intervals of is the Euclidean topology.
The members of are called the open sets of ; the complements of the open sets are called closed. If , then is a topology on , called the induced topology. For instance, the interval is closed in the Euclidean topology of , as it is the complement of the open set ; and is an open subset of .
An open cover of is a family of open sets of whose union is . Open covers can be thought of as models of the level of detail one can achieve by observing a space while one’s vision is, for some reason, blurred: an observer will be able to tell point from point if and only if there are two disjoint open sets from the given open cover (possibly, the whole topology) one of which contains and the other .
A topological space is compact if every open cover contains a finite subcover. Suppose that, to compensate our blurred vision, we hire guardians to keep track of the points of the space, but the contract with the security company states that each guardian will only watch the points of a given open set: compactness then ensures that we can always watch the whole space while only hiring finitely many guardians. (This similitude was suggested to me by Tommaso Toffoli.) The Heine-Borel theorem states that every open cover of the closed and bounded interval contains a finite subcover. Tychonoff’s theorem implies that the set of infinite words on a finite alphabet , whose open sets are unions of sets of the form with , is a compact space.
A function between two topological spaces is said to be continuous if the counterimage of any open subset of the codomain is an open subset of the domain. According to our similitude, going back along a continuous function does not worsen the level of detail: if the images of two points can be distinguished in the codomain, then the original points can also be distinguished in the domain. The usual formulation of continuity from Calculus courses is, in fact, a special case of this definition.
If is a compact space, then for every open cover of there is a minimum number of elements of a finite subcover of . We set , where the logarithm is taken in a fixed base .
Let and be open covers of : we say that is a refinement of if for every there exists such that : we then write . The intuitive meaning is that allows a level of detail no smaller than does: if allows telling any two points apart, so does .
It is immediate to check that is a preorder: usually, is not antisymmetric. As a counterexample, let:
- ; and
Then and , but The example above can be modified to show that the case is also possible.
The common refinement of and is
Clearly, ; if is any open cover such that , and , then there exist and such that : thus, too.
The operation is commutative and associative, but usually not idempotent: as a counterexample, if and , then . Observe that, with our choice, : this is actually a general fact.
Theorem. Let be a compact topological space, let and be open covers of , and let be a continuous function.
- If and then .
- If then and .
- If then .
- . If is surjective then equality holds.
Given an open cover of a compact space and a continuous function , let
Then for every ,
so that : by Fekete’s lemma,
exists. The quantity above is called the entropy of relative to . The topological entropy of the continuous function is then defined as
Intuitively, topological entropy is the worst-case amount of average
information per iteration required to reconstruct the initial state, given the
current one. As a quick sanity check, let us verify that the identity has zero topological entropy. Clearly, whatever is, so that for every , and whatever is.