# 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