Re: -- quasi-paths vs connected ordered spaces

On 31 Sty, 04:32, quasi <qu...@xxxxxxxx> wrote:
On Wed, 30 Jan 2008 17:42:43 -0800 (PST), Zbigniew Karno



<zbigniew.ka...@xxxxx> wrote:
On Jan 29, 2:54 pm, quasi <qu...@xxxxxxxx> wrote:
Let X be a topological space.

Given distinct points a,b in X, call a subset P of X a quasi-path from
a to b if

(1) a,b are in P.
(2) P is connected.
(3) No proper subset of P satisfies (1) and (2).

If X is a continuum (i.e. compact & connected space),
then such P is called irreducible subcontinuum of X.

Question:

Must a quasi-path from a to b be homeomorphic to a connected ordered
space?

quasi

For a simple counterexample consider X to be
a warsaw circle, i.e. X = S cup L, where S is
a circle in the complex plane and L is a spiral
approximating S from the inside starting at some
point p. In particular, the closure of L = X in X,
and X is connected.
Now if s belongs to S, then the irreducible
subcontinuum of X, which contains both p and s,
coincides with X. But X is not homeomorphic to
a connected ordered space, because X contains
the circle S.

Hmmm ...

I don't see why X is a minimal connected set containing p and s. Since
I don't require such sets to be compact, it seems that (L union {s})
is a smaller such set -- in fact, the unique minimal such set. Also,
it's easily seen that (L union {s}) can be ordered, consistent with
its topology.

What am I missing?

quasi


I am missing something, fixing on a compact case.
So, this example works if in the definition the
compactness is required.

--
Z. Karno
.

Relevant Pages

  • Re: -- quasi-paths vs connected ordered spaces
    ... Given distinct points a,b in X, call a subset P of X a quasi-path from ... then such P is called irreducible subcontinuum of X. ... a circle in the complex plane and L is a spiral ...
    (sci.math)
  • Re: identification map
    ... as (0,1] is not compact, and the circle is ... to show not an identification map. ... sorry about this confusion. ...
    (sci.math)
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