Speaker: Sakae Fuchino (Kobe Univ.)
Title: Dow's metrization theorem and beyond
Time/Date: 4:30-6:00pm, Wednesday, January 8, 2014
Room: 122 Math. Sci. Building
Abstract: Alan Dow's metrization theorem of 1988 asserts: if a countably compact topological space X is such that all subspaces Y of X of cardinality ≤ \aleph_1 are metrizable then X itself is also metrizable.
This theorem was one of the first example of "mathematical" theorems (in the usual axiom system ZFC of set theory without any other additional axioms) whose optimal proof uses the method of elementary submodels in a very essential way.
This type of assertions where "a certain property of some large structure is attributed to the same property of all small substructures" (or in contraposition: "if a large structure does not have certain property then there is a small substructure which does not satisfy this property") is often called "reflection theorems".
If the countable compactness of the space is dropped totally from Dow's reflection theorem then the assertion is simply wrong.
On the other hand, if the countable compactness of the space is replaced by local countable compactness then the assertion becomes independent from ZFC (for the proof of consistency of the assertion, we need the existence of a fairly large large cardinal).
I found many characterizations of this last reflection theorem on metrizability for spaces with local countable compactness in terms of reflection of many other mathematical notions like countable coloring number of graphs, left separatedness of topological spaces, openly generatedness of Boolean algebras, etc. in recent joint works with Hiroshi Sakai, Asaf Rinot, Lajos Soukup and Toshimichi Usuba.
In the talk, I shall give a more detailed description of these and other related results together with a quick review of the background knowledge of set theory and logic needed to obtain a (rough) understanding of the significance of these results.