Re: Uncountable closed subspaces of a set of aleph_1 with minimal well-ordering are homeomorphic
- From: William Elliot <marsh@xxxxxxxxxxxxxxxxxx>
- Date: Mon, 29 Jan 2007 02:28:02 -0800
From: BP <bp1099@xxxxxxxxxxx>
Let A be a set of cardinality aleph_1 with minimal well-ordering <=
and a topology T generated by this well-ordering. Show that every
uncountable closed subspace Y of A is homeomorphic to A .
I previously asked this question on Ask a Topologist about a week
and a half ago. A member there resoponded and posted the following
here: http://groups.google.com/group/sci.math/browse_thread/thread/
2e9d4e1711dd4bd2/e3137947a09047f3
This link doesn't open from within the post because you, your word
processor or your transmission routers inserted a carrage return. I'll
repost the link here without carrage return to experment if my
transmission routers insert a carrage return or not.
http://groups.google.com/group/sci.math/browse_thread/thread/2e9d4e1711dd4bd2/e3137947a09047f3
Use of outside links is not recommended for newsgroups, especially links
for the ramblings of a thread.
Now I'm dealing with cardinals rather than ordinals, so I'm not
sure that you can do what he did here. I have been unable to prove
this and it would be nice to see what the proof looks like.
Cardinals are initial ordinals. If you followed the discussion, it
concluded your problem is the same as showing an unbounded (equivallently
uncountable) closed subset A of omega_1 is homeomorphic to omega_1. It
was determinded that A was order isomorphic to omega_1. Thence the
discussion resulted in determining if the instrinstic order topology of A
was the same as the subspace topology of A. This is the situation, not
because of what was discussed (which was incomplete), but because
If K is a closed subset of a bounded complete (Dedekind complete) linear
order (S,<=), then the subspace topology of K, (S,<=)|K is the same as the
instrinstic linear order topology of K, (K, <= /\ KxK).
The full proof of this including the exact conditions when the subspace
topology is the instrinstic subset topology, is outlined in over a page of
my terse densily packed notes.
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