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,

http://ocw.mit.edu/courses/mathematics/18-965-geometry-of-manifolds-fall-2004/lecture-notes/lecture2.pdf

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 .”

Set and

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

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