Suslin sets and projections

Question

In a paper I am reading about optimal transport (we are in a proper metric space, https://arxiv.org/pdf/2004.08934.pdf, page 49 at the end), there's written in a proof "Being (this set) the projection of a Borel set, then it is Suslin".

I am finding hard to use the definitions of "Suslin sets" to get why this holds true. Can someone give me why what the authors wrote is true and some references where I can find properties and results about Suslin and/or Borel sets?

Also, why the sets in that page ($\mathfrak{a}(\mathcal{T}^e_V)$ and $\mathfrak{b}(\mathcal{T}^e_V)$) are co-Suslin?

For Borel sets and the Borel hierarchy (preliminaries to Suslin sets), see most any item in this answer. Of those items, Measure Theory by Cohn has a fair amount on Suslin sets. Besides those books (and maybe better general references in general for your needs), see Set Theory by Abhijit Dasgupta and A Course on Borel Sets by S. M. Srivastava and Classical Descriptive Set Theory by Alexander S. Kechris. — Dave L. Renfro, Sep 23 '21 at 21:02
Also: Glen Alan Schlee, On the Development of Descriptive Set Theory, Master of Arts Thesis (under Richard Daniel Mauldin), University of North Texas (Denton, Texas), August 1988, iii + 66 pages. — Dave L. Renfro, Sep 23 '21 at 21:06

score 1 · Answer 1 · answered Dec 25 '22 at 07:41

1

In general, all Borel sets are Suslin sets, and all images of Suslin sets under continuous functions are also Suslin sets. These facts are proved in theorems 1.6.9 and 1.6.12 in the book “Geometric Integration Theory” by Krantz and Parks.

answered Dec 25 '22 at 07:41

Cheuk Hwang

125
4

Suslin sets and projections

1 Answers1