This originally was sent to stackexchange:
This question is prompted while I was working through the MIT OCW (Massachusetts Institute of Technology, Open CourseWare) for 18.965, “Geometry of Manifolds,” in its Lecture 2,
Near verbatim, the setup for the example of a long line from there is this:
Let denote the smallest uncountable totally ordered set.
Consider the product with dictionary order topology.
Give charts as follows.
if , let and
If , “let denote the successor of .”
My questions are the following: what is the domain for that second chart, ? It’s unclear to me if this is a union or what’s going on with the supremum for .
Also, in this context, could someone give an example of what it means to be a successor for , to clarify what it means, and in general?
Thanks, and for those working through MIT OCW 18.965 or are seriously interested in learning differential topology online, I’m keeping a blog with my progress and some, hopefully helpful, thoughts, discussions, and solutions at ernestyalumni.wordpress.com.
Mrowka, Tomasz. 18.965 Geometry of Manifolds, Fall 2004. (MIT OpenCourseWare: Massachusetts Institute of Technology), http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004} (Accessed 29 Nov, 2014). License: Creative Commons BY-NC-SA