>
> "mimus" hotmail.com> wrote in message
> news:uYCdncq7zKAt1bXVnZ2dnUVZ_rHinZ2d@giganews.com...

>> On Mon, 12 May 2008 23:25:47 +1000, Peter Webb wrote:
>>

>>> "mimus" hotmail.com> wrote in message
>>> news:WZGdnYk6ysIsorXVnZ2dnUVZ_qHinZ2d@giganews.com...
>>>

>>>> On Mon, 12 May 2008 18:36:20 +1000, Peter Webb wrote:
>>>>

>>>>> "mimus" hotmail.com> wrote in message
>>>>> news:8uCdnZuH0sBCzIDVnZ2dnUVZ_hKdnZ2d@giganews.com...
>>>>>

>>>>>> On Sat, 03 May 2008 21:33:59 -0500, Tim Weaver wrote:
>>>>>>

>>>>>>> mimus wrote:
>>>>>>>

>>>>>>>> At least, I swept up at least twice as much glass as could possibly
>>>>>>>> have
>>>>>>>> been in the original.
>>>>>>>>
>>>>>>>> And that's only possible if you do an infinite decomposition of the
>>>>>>>> object, as exemplified by the Tarski-Banach ball.
>>>>>>>>
>>>>>>>> Maybe this is a sign I should mop my kitchen-floor.
>>>>>>>>
>>>>>>>> Will I need an infinite mop?

>>>>>>>
>>>>>>> Yes, if you have a mobius shaped floor.

>>>>>>
>>>>>> Mobius strips are usually finite.
>>>>>>
>>>>>> Just unbounded.

>>>>>
>>>>> Technical note: Mobius strips have a single boundary.

>>>>
>>>> ok fine.
>>>>
>>>> (As soon as I read that, my head tried to encompass an unbounded Moebius
>>>> strip and couldn't do it.)

>>>
>>> Well ...
>>>
>>> Imagine the width of the Mobius strip was infinite. You would then have a
>>> surface that was unbounded in both directions, finite in one direction
>>> and
>>> infinite in the other - sort of like a cylinder, but its not. Nor is it a
>>> Klein bottle or cross-cap. It can't be embedded in R^3 as it self
>>> intersects. But it is a reasonable interpretation of an "unbounded Mobius
>>> strip", whatever its real name is (if it has one).

>>
>>
>>
>> A single-sided infinite plane or saddle or a single-sided sphere, is
>> what it looks to me like what we're lookin' at, yes it does. Yes.

>
> No. A single sided infinite plane is infinite in both directions, a single
> sided sphere is finite in both directions. So my "unbounded Mobius strip" is
> not isomarphic to either of these.

>> I think Klein bottles cheat with that penetration business-- tearing is a
>> no-no in algebraic topology,

>
> No tearing is neccessary. You simply lift the "neck" of the bottle into the
> fourth dimension, move it "over" the outside surface, then reconnect it.
>
> The problem you are having is that a Klein bottle, despite being a perfectly
> reasonable 2D surface, can't be embedded into R^3. (And nor can my infinite
> Mobius strip). However, it is only your parochial desire that 2D surfaces
> should be embeddable in R^3, there is no reason that they should be - they
> are both embeddable in R^4 without any problem, and there is no reason to
> limit the consideration of 2D surfaces to those which can be constructed in
> 3D, even if you think you live in 3D world.