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Many concepts in General Topology are the direct abstraction of very profound and natural concepts (think of structures as topology or uniformity themselves, separation axioms, quotient and identification topologies, connection, compactness...)

I am having hard times trying to understand the fundamental idea behind Covering Axioms, namely concepts as paracompactness and the various types of refinements (point and nbd finite, starring, barycentric). I understand the definitions and I know their importance with respect to metrizability and other procedures such as partitions of unity, but can't figure them out, and they seem to me rather tachnical. I have tried to see them as a generalization of compactness but this does not seem very fruitful. So I ask:

-Do Covering Axioms have a direct interpretation, or are they just a technical means to other constructions?

-What is the extent of their relevance?

-Is there a direct "categorical" link with ordinals, and order concepts, which appear very frequently in proofs?

YuiTo Cheng
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Francesco Bilotta
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    Seemingly "covering axioms" is too broad... Could you pin down what topological invariants you don't understand? I noticed you've added a "paracompactness" tag. Does this mean you have particular difficulty understanding it? – YuiTo Cheng Apr 06 '19 at 11:07
  • By covering axioms I intend in general those properties regarding covering charachterization of a space, like paracompactness, first and second countable space, I used this term as it is the one used in my textbook (Dugundji). Similarly to when, for separation axioms, one usually intends T0, T1,T2,T3 and T4 spaces and so on. Indeed paracompactness is the most important case, it seems to me, so some intuition about it would certainly be appreciated! – Francesco Bilotta Apr 06 '19 at 13:28
  • As a comment, it might be an illusion of your present point of view that "ordinals, and order concepts ... appear very frequently in proofs". Although this may be true in other branches of mathematics (such as mathematical logic), this is not really the case in topology. – Lee Mosher Apr 06 '19 at 15:04
  • @LeeMosher Yes, I mean, very frequently in proofs about covering and refinements, not in general topology as a whole. By the way, since I am somehow curious about the point you made: the main area where I saw proofs involving directly such concepts as well orderings, Zorn's Lemma and so on, apart from Algebra, is (Functional) Analysis. In my vision, Topology is something foundational, whereas Analysis is more "high level" language: why something so basal as the Axiom of Choice has more direct use in Analysis than Topology? – Francesco Bilotta Apr 07 '19 at 09:47
  • @FrancescoBilotta One reason why is because analysis involves a lot of work with measurable spaces/measurable functions, which are in that context "coarser"/"lower level" structures than the topological structures that induce them. For example, the axiom of choice is necessary to define unmeasurable sets, cf. https://en.wikipedia.org/wiki/Solovay_model Really heuristically, topology is often used so we can "smooth out considerations involving individual points" -- if you don't need to consider problems arising from individual points, then you don't axiom of choice to describe problems. – Chill2Macht Feb 06 '22 at 13:50

1 Answers1

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The most useful and simplest covering axiom is compactness: every open cover has a finite subcover. This has many applications and is well-behaved wrt topological operations like products.

Classical are also the "cardinality variations":

  • Lindelöf : every open cover has a countable subcover.
  • Countable compactness: every countable open cover has a finite subcover.

And this generalises easily to all sorts of $(\kappa,\lambda)$-compactness for cardinals $\lambda \le \kappa$: every open cover of size at most $\kappa$ has a subcover of size $< \lambda$, so that compactness becomes $(\infty,\aleph_0)$-compact, countably compact becomes $(\aleph_0, \aleph_0)$-compact and Lindelöf $(\infty, \aleph_1)$-compact (with the convention that $\infty$ means no restriction on cardinality at all). One can now study preservation of these by products of various sizes, etc.

Another way to vary is to consider refinements instead of subcovers. But the property that every open cover has a finite refinement (easily) turns out to be equivalent to compactness. The property that every finite open cover has a finite closed refinement (that is also a cover!) is (almost?) equivalent to being $T_4$, so we are getting non-trivial properties, close to normality. Instead of finite we can ask for point-finite (every point of $X$ is in at most finitely many members of the cover) or locally finite (every point has a neighbourhood that intersects at most finitely members of the cover, this implies point-finite of course), or even star finite (every member of the cover only intersects at most finitely many other members of the cover, this implies locally finite for open covers). Here we do need to consider refinements instead of subcovers:

$X$ is compact iff every open cover has a point-finite subcover. (see Brian's answer for a proof)

So we get a possibly new property by demanding that every open cover has a locally finite (or point-finite, or star-finite) open refinement. This is the famous paracompactness property (or weak paracompactness/metacompactness for point-finite and strong paracompactness for star-finite refinements). It has the nice property that paracompactness plus Hausdorff implies normal (as with compactness) and that is preserved by closed continuous maps (Michael's theorem). It's not very well-behaved under products but Stone showed that all metric spaces are paracompact, so there are many non-compact examples.

Further seeming weakenings or variations of paracompactness turn out to be often equivalent to paracompactness: $X$ is paracompact iff every open cover has a $\sigma$-locally finite open refinement, or a locally finite closed refinement etc. See my notes here and here and here e.g. So paracompactness is a nice middle ground with the most theorems (like the existence of partitions of unity which also turns out to be equivalent).

It's not real intuition as such, but more a motivation how this property has been defined after a lot of related results in metrisation theorems that showed the usefulness of working with locally finite families and refinements of covers (Bing-Nagata-Smirnov etc.). The Handbook of Set-Theoretic Topology (from 1984) has a whole chapter on more properties like this (further weakenings that are not equivalent any more) and their relations. Not a lot of research is done any more in these, is my impression.

As to your last question: ordinals often not paracompact spaces in their order topology: $\omega_1$ (the first uncountable ordinal) is a standard example space that is not paracompact in the order topology e.g.

Henno Brandsma
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