Dual Ramsey, an introduction – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Dual Ramsey, the Gurarij space and the Poulsen simplex 1 (of 3).

Lecturer: Dana Bartošová.

Date: December 12, 2016.

Main Topics: Comparison of various Fraïssé settings, metric Fraïssé definitions and properties, KPT of metric structures, Thick sets

Definitions: continuous logic, metric Fraïssé properties, NAP (near amalgamation property), PP (Polish Property), ARP (Approximate Ramsey Property), Thick, Thick partition regular.

Lecture 1 – Lecture 2 – Lecture 3

Ramsey DocCourse Prague 2016 Index of lectures.

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Facts about the Urysohn Space – Some useful, some cool

(This is almost verbatim the talk I gave recently (Feb 23, 2012) at the Toronto Student Set Theory and Topology Seminar. I will be giving this talk again on April 5, 2012)

I have been working on a problem involving the Urysohn space recently, and I figured that I should fill people in with the basic facts and techniques involved in this space. I will give some useful facts, a key technique and 3 cool facts. First, the definition!

Definition: A metric space U has the Urysohn property if

  • U is complete and separable
  • U contains every separable metric space as an isometric copy.
  • U is ultrahomogeneous in the sense that if A,B are finite, isometric subspaces of U then there is an automorphism of U that takes A to B .

You might already know a space that satisfies the first two properties – The Hilbert cube [0,1]^\omega or C[0,1] the continuous functions from [0,1] to [0,1] . However, these spaces are not ultrahomogeneous. Should a Urysohn space even exist? It does, but the construction isn’t particularly illuminating so I will skip it.

Continue reading Facts about the Urysohn Space – Some useful, some cool