{"id":1591,"date":"2018-12-25T19:11:34","date_gmt":"2018-12-25T18:11:34","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1591"},"modified":"2022-11-19T15:09:29","modified_gmt":"2022-11-19T14:09:29","slug":"on-countability","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1591","title":{"rendered":"On countability"},"content":{"rendered":"

Everyone has its own prejudices.\u00a0 One of mine is that countability assumptions are usually superfluous.\u00a0 E.g., instead of defining convergence with sequences, and then realizing that closed sets can only be characterized by sequences in first-countable<\/em> spaces, it is better to define convergence using nets, or using filters\u2014that works in every space.<\/p>\n

However, there are some places in mathematics where you really need countability assumptions, and some of them are somehow unexpected.<\/p>\n

The expected places<\/h2>\n

You would expect countability to be important in questions related to \u211a or \u211d.\u00a0 Metric spaces are first-countable, and that is because metrics take their values in \u211d+<\/sub> \u222a {\u221e}.\u00a0 Measures, which also take their values in\u00a0\u211d+<\/sub> \u222a {\u221e}, are \u03c3-additive, i.e., they map countable disjoint unions to sums, for some pretty good reasons, too, related to the existence of non-measurable sets, and results such as the monotone convergence theorem, which applies to\u00a0countable<\/em> sequences of maps to be integrated.<\/p>\n

But there are some places where countability is important, and neither \u211a or \u211d is involved.<\/p>\n

Projective limits<\/h2>\n

I have spent quite some time on projective limits already.\u00a0 Remember: the projective limit of a projective system of non-empty topological spaces may<\/em> be empty.\u00a0 It is non-empty if all the spaces involved are compact and sober\u2014this is part II<\/a> of my previous series posts on Steenrod’s Theorem\u2014or if the system has a cofinal countable<\/em><\/a> subsystem.\u00a0 And we have seen one counterexample, which indeed has no cofinal countable subsystem.<\/p>\n

Note that countability is not required<\/em>. \u00a0What is required is countability\u00a0or<\/em> the fact that all spaces are compact and sober. \u00a0All the examples that we shall see will be of this form: we will need either a pretty strong topological condition (here, compactness and sobriety), or a countability condition.<\/p>\n

The Plotkin powerdomain<\/h2>\n

The Plotkin powerdomain was introduced so as to give semantics to non-deterministic choice in semantics. \u00a0I will not describe what this is, because this is a pretty complicated subject (see Section 6.2.1 of [1]), and I will describe a better example, where countability makes wonders, in the next section.<\/p>\n

Let me just say that this is one of the first examples I saw where countability (or<\/em> coherence) was required. \u00a0To wit,\u00a0Theorem 6.2.19 of [1] states that the Plotkin powerdomain of a continuous dcpo\u00a0X<\/em>, which is defined there as a certain continuous dcpo (by construction, as the rounded ideal completion of an abstract basis), is isomorphic to a space of so-called lenses over\u00a0X<\/em>, provided that\u00a0X<\/em> is\u00a0\u03c9-continuous, not just continuous. \u00a0This is where the countability condition comes into play.<\/p>\n

Again, countability is not really needed. \u00a0Instead, if X<\/em> is not only a continuous dcpo, but also a coherent one, then the Plotkin powerdomain is also isomorphic to the space of lenses (Theorem 6.2.22 of [1]; see also Proposition 4.45 of [3], and its corrigendum, for a complete proof that the space of lenses is a continuous dcpo in this case). \u00a0If we do not wish to require coherence, countability in the form of\u00a0\u03c9-continuity is necessary, as the pretty sophisticated construction of\u00a0Exercise 6.2.23(11) of [1] shows.<\/p>\n

The Smyth powerdomain<\/h2>\n

The Smyth powerdomain\u00a0Q<\/strong>(X<\/em>) is the set of compact saturated subsets of\u00a0X<\/em>. \u00a0(Usually the empty set is excluded. \u00a0I will not do this.) \u00a0It can be equipped with either one of two classical topologies:<\/p>\n