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Bookmark or Share- general topology - Regarding the connectedness of the Long Line ...I believe that the Long Line is a mapping from the real line to the real line such that each point on the Long Line represents a copy of the real line. This doesn't quite parse for me, but it sounds like you're talking about a particular topology on $\mathbb{R}\times\mathbb{R}$ - thought of as replacing each point in $\mathbb{R}$ by its own ...
- general topology - Fundamental Group of the Long Line - Mathematics ...The key to understanding questions like this about the long line is the following observation. Lemma ...
- Can the definition of "The Long Line" be clarified?A better (IMO) description of the long line is $[0,\Omega) \times [0,1)$ ordered lexicographically: $$(\alpha, t) \le_L (\beta, u) \iff (\alpha
- general topology - Is there such a thing as the *longer* line ...$\begingroup$ The problem is that any such attempt will lose the homogeneity so important to the long line itself: the point corresponding to $\omega_1$ will not have a neighborhood isomorphic to $\mathbb{R}$ (more snappily, lots of ordinals below $\omega_2$ have uncountable cofinality, and so any long line built off of $\omega_2$ will fail to be first-countable).
- general topology - Prove the long line is not contractible ...Define the long ray to be the ordered set $\omega_1 \times [0,1)$ taken in lexicographical order. As a space, it is given the order topology. Define the long line to be the space obtained by gluing together two long rays together at their initial points. Prove the long line is not contractible. An outline of a proof can be given as follows:
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