Re: Moebius Band is not homeomorphic with a Torus

Narasimham wrote:
David Marcus wrote:
Narasimham wrote:
David Marcus wrote:
What does the word "homeomorphism" mean? Please define it and give an
example.

Repeating this again, the normally accepted text-book definition.

One to one mappings that stretch and bend their domains into their
ranges without tearing are called homeomorphisms.

Incorrect. For a map to be a homeomorphism, both the map and its inverse
must be one-to-one and continuous.

Really ? and what if there is no inverse possible?..please see below.

( pp39, Maynard J. Mansfield, Introduction to Topology, Princeton, NJ,
D Van Nostrand)

Are you sure you are quoting the book correctly?

Yes. (You can also check it out). As it is an introductory book, a
2-way continuity may have been implied without prejudice to the
context. But the flexible rubber membrane intuition comes through very
strongly in many texts.

There are essentially three components for a homeomorphism for 2-domain
parameter (u,v) function mapping into a 3-space range parameters
(x,y,z).

1) Parameter domain space, two elements (u,v)
2) Parameter range space,three elements (x,y,z)
3) Functional dependences, three relations x(u,v) = fx(u,v), y(u,v) =
fy(u,v), z(u,v) = fz(u,v).

So, f: R^2 -> R3.

Yes.

[ u is short for u1 <= u <= u2 etc.]. plot3d,ParametricPlot3D surfaces
of Maple or Mathematica are standard examples of homeomorphisms.

The map f will only be a homeomorphism if f is continuous and f^{-1} is
continuous.

May be satisfied in case of my tori set, but I am not too sure what
inverse f should mean here. f: R^2->R^2 both ways say in [{x = u+ v
,y = u-v} -> {u= (x+y)/2 ,v = (x-y)/2}] is possible but for f:
R^2->R^3 say in forward direction in [ {x = u+ v ,y = u-v, z =u*v }]
is possible but in reverse / inverse f: R^3->R^2 clearly (x,y,z)
cannot be evaluated in terms of {u.v}]. Similarly f: R^2-> R^4 is OK
but in f: R^4-> R^2 inverse function is just not parametrically
possible.

If it is to be seen in this light,an insistence on f^{-1} between
parameters of unequal space dimension makes no sense. But please
comment.

The inverse only has to be defined on the image of the function, not on
all of R^3. In your example of an f: R^2 -> R^3, you have a set D =
{(u,v) | u1 <= u <= u2, v1 <= v <= v2 } that f is defined on. It is only
f restricted to D that needs to have an inverse. The inverse will be
defined on just f(D), which is a subset of R^3.

Many
examples can be given, for example the catenoid and helicoid with
isometry and uniform dilatation:

{ u cos(t), u sin(t),t} , (0 < t < 2 pi), ( -1 < u < 1) and
{ u cos(t), u sin(t),arccosh(u) } , (0 < t < 2 pi}, ( -1 < u < 1)

Topology is indifferent to number of plot points m in a curve. It
is__not__ a relevant parameter. OK?

I don't know what you mean by "number of plot points".

I was referring to the Mathematica plotting option.

But independently we may say to convey the same: In f:R^2->R^2 the set
of polygons inscribed in a circle (diangle,triangle,quadrangle or
quadrilateral, pentagon, .. octagon) are all homotopic and homeomorphic
to the circle. And so between any two of them.

Even if m is a relevant parameter _____

Embedding of (u,v,m) maps into R^3 or (x,y,z) should (IMHO) should
yield a torus set as I indicated homeomorphic between Moebius Band and
other tori agreeably in a somewhat constricted sense, but that should
be applicable to ALL elements of the torus set, not just to the m = 1
MB case alone. Is it not?

I don't know what you mean by this paragraph. "Embedding of maps"?
"Torus set"?

That was in some error. I wanted to say: (u,v,m) maps into R^3 with
(x,y,z) etc.The torus set was pictured in first post ( MB, square
tube,...ridged torus,torus for m= 1,2,6, Infinity)

However, there should be no disagreement about the higher dimension
embeddings:

Embedding of (u,v,m) maps into R^4 or (w,x,y,z) yields a free torus
superset homeomorphism between Moebius Band and torus. As sufficient
(one extra degree of freedom) accomodation is available.

What is the confusion here? Why is not the central point (Topic: When
is a homotopy not a homeomorphism?) simply stated topologically and
discussed ?

A homotopy is never a homeomorphism. Homotopies are between maps.
Homeomorphisms are between spaces.

Oh, but is not homotopy itself a map?

Yes.

created by variation of a single
parameter?

Yes.

a collective/group desription?

Not sure what this means, but I think the answer is no. A typical use of
homotopy is to show two spaces are not homeomorphic. For example, in R^
2, all circles are homotpic to a point, but in R^2\{(0,0)}, a circle
that loops around the origin is not homotopic to a point.

--
David Marcus
.

Relevant Pages

  • Re: Moebius Band is not homeomorphic with a Torus
    ... For a map to be a homeomorphism, both the map and its inverse ... be applicable to ALL elements of the torus set, not just to the m = 1 ... A homotopy is never a homeomorphism. ...
    (sci.math)
  • Re: Moebius Band is not homeomorphic with a Torus
    ... One to one mappings that stretch and bend their domains into their ... For a map to be a homeomorphism, both the map and its inverse ... be applicable to ALL elements of the torus set, not just to the m = 1 ...
    (sci.math)
  • Re: Moebius Band is not homeomorphic with a Torus
    ... other tori agreeably in a somewhat constricted sense, ... be applicable to ALL elements of the torus set, not just to the m = 1 ... things: homeomorphism and homotopy? ... get an explicit formula depends on your map. ...
    (sci.math)
  • Re: Moebius Band is not homeomorphic with a Torus
    ... By plotting the set of cardioids: ... portions (assuming the procedure is valid for homotopy) of all ... There are essentially three components for a homeomorphism for 2- ... be applicable to ALL elements of the torus set, not just to the m = 1 ...
    (sci.math)
  • Re: Moebius Band is not homeomorphic with a Torus
    ... By plotting the set of cardioids: ... portions (assuming the procedure is valid for homotopy) of all ... There are essentially three components for a homeomorphism for 2- ... be applicable to ALL elements of the torus set, not just to the m = 1 ...
    (sci.math)
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