[FOM] The "long line".
joeshipman at aol.com
Fri Jun 22 17:33:11 EDT 2007
2) Yes for any ordinal less than or equal to aleph_1; but "the" long
line is the one which uses aleph_1 -- with larger ordinals the points
above aleph_1 are essentially different than the ones below.
One of the questions I used to like to ask students is "Define a
1-manifold as a nonempty Hausdorff space in which each point contains a
neighborhood homeomorphic to (0,1). How many nonisomorphic connected
1-manifolds are there?" They would always answer two (the real line and
the circle) but of course you have to include the long line (without
the 0 point) and the doubly long line also. This is why textbooks often
require manifolds to have additional properties such as paracompactness
or metrizability or having a countable basis.
What is carefully de-emphasized is that this means "being a manifold"
is not really a local property that can be defined in terms of
properties of neighborhoods of points. All definitions of "manifold"
that exclude the long line are necessarily "global".
Can anyone think of any other properties of topological spaces which
appear to be local but actually are not?
From: Bill Taylor <W.Taylor at math.canterbury.ac.nz>
To: fom at cs.nyu.edu
Sent: Thu, 21 Jun 2007 11:44 pm
Subject: [FOM] The "long line".
IIRC, the long line is obtained by starting with a large ordinal,
(say aleph_1), and inserting a copy of interval (0,1) after each
(and extending the order relation in the obvious way).
(i) If the starting ordinal is a countable one, is the final result
order-isomorphic to [0,1) ?
(ii) If the long line is preceded by a reversed long line, and the two
zero-points identified, is the resulting ordered set automorphic
with any other point is mappable to the double-zero?
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