Long Line | search
Click here to view Long Line news from 60+ newspapers.
Bookmark or Share- 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:
- general topology - The Long Line is not second countable - Mathematics ...The construction of the Long Line topology. 9. Are countably infinite, compact, Hausdorff spaces ...
- general topology - Extended Long Line isn't path-connected; proof ...Since the long line is Hausdorff, there is an arc from a to b. Show the arc is increasing and use that to construct an order embedding of omega_1 + 1 into [0,1], an impossibility.
- general topology - Prove long line is path-connected. - Mathematics ...$\begingroup$ Also see this note, of which you should only not trust the last statement: the long line is normal (despite being non-paracompact, but it is countably paracompact). The fact that the initial segments are homeomorphic to open intervals in $\Bbb R$, as shown, implies path-connectedness.
- The construction of the Long Line topology - Mathematics Stack ExchangeIn the most usual formulation of set theory, $\omega_1$ is the set of all countable ordinals; so the two definitions of the long line are equivalent. Squeezing an interval between each countable ordinal is equivalent to turning each countable ordinal into a half-open interval, and the set you get is the union of $\omega_1$-many disjoint half ...
More
