charts for an atlas for the long line (cf. stackexchange)

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 S_{\Omega} denote the smallest uncountable totally ordered set.

Consider the product X = S_{\Omega} \times (0,1] with dictionary order topology.

Give X charts as follows.

\forall \, (\omega, t) \in X, \\
if t\neq 1, let U_{(\omega,t)} = \lbrace \omega \rbrace \times (0,1) and

\begin{aligned}                                                                                                  & \phi_{(\omega,t)}: U \to \mathbb{R} \\                                                                      & \phi_{(\omega,t)}(\omega,t) = t                                                                               \end{aligned}

If t=1, “let S(\omega) denote the successor of \omega.”

Set U_{(\omega,1)} = \lbrace \omega \rbrace \times (0,1]\text{sup}\lbrace S(\omega) \rbrace \times (0,1) and

\phi_{(\omega,t)}(\eta, t) = \begin{cases} t & \text{ if } \eta = \omega \\    t + 1 & \text{ if } \eta = S(\omega) \end{cases}

My questions are the following: what is the domain for that second chart, (\phi_{(\omega,t)}, U_{(\omega,1)}\  )? It’s unclear to me if this is a union or what’s going on with the supremum for S(\omega).

Also, in this context, could someone give an example of what it means to be a successor for \omega, 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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