Re: Is CW a local property?

On 6 Okt., 11:51, Keith Ramsay <kram...@xxxxxxx> wrote:
On Oct 5, 4:04 am, sanchopanch...@xxxxxx wrote:
|can anyone help me to show that the Long-Line is not a CW complex but
|a 'local CW complex' in the sense of the first post?

Since each point on the long line has an
open neighborhood homeomorphic to an open
interval on the real line, the latter is
easy.

The long line doesn't have a homeomorphic
copy of the interior of B^n for n>1, so if
it were a CW complex, it would be a 1-frame.
Suppose it is. The points in the 0-frame
can't have an upper bound because there
would be no way to glue a 1-cell so that it
included any of the points above the upper
bound. So there have to be uncountably many
points in the 0-frame. It's only possible
to connect a point in the 0-frame to two
others, so only countably many points in the
0-frame can be path-connected. But the long
line is path-connected. Details need to be
filled in.

Keith Ramsay


Hello,
thank you! The long line isn't a compact space. Is the question above
trivial for compact spaces?
Greets
Sancho

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