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Topic: Connected and Locally Connected
Replies: 4   Last Post: May 20, 2014 11:32 PM

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Roland Franzius

Posts: 433
Registered: 12/7/04
Re: Connected and Locally Connected
Posted: May 20, 2014 2:24 AM
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Am 20.05.2014 06:50, schrieb William Elliot:
> On Mon, 19 May 2014, Roland Franzius wrote:
>> Am 19.05.2014 10:46, schrieb William Elliot:
>
>>> A space S is a Peano space when S is compact, connected,
>>> locally connected and metrizable. Peano spaces are path,
>>> in fact, arc connected. Is there a connected, locally
>>> connected and metrizable space that's not path connected?

>>
>> Oh I see, you really want to discuss with the Mückenheim corona something
>> like
>>
>> In this paper we prove that if a semilocally 1-connected Peano space X
>> admits a positively expansive open map ?, then X is homeomorphic to an
>> infra-nil-manifold and ? is topologically conjugate to an expanding
>> infra-nil-endomorphism?

>
> Is that an example, a counter example or a speculation?a
>


Since the encyclopedia of mathematics does not have as a lemma Peano
spaces, I tried a scholar.google search for "peano spaces", the term
that was more significant than the rather broad terms "locally" and
"connected/pathconnected", that bring only those standard exercises from
topology lecture notes like "topologist's comb" and "topologist's sine".

That cited abstract is indeed from paper and reminded me of those happy
times when Bourbaki topologists were paid by the NATO research founds
and created new categorial definitions by tons per year.

The article is

Positively expansive open maps of Peano spaces
K Hiraide - Topology and its Applications, 1990 - Elsevier

but I don't read such stuff, always remembering the warning from Reed/Simon:

Functional analysis (and operator algebra and group theory), the
paradigma of quantum mathematical physics, is no frutful playground for
topologists.

--

Roland Franzius



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