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Topology


Date: 12/30/97 at 18:27:12
From: Kristin Bullard
Subject: Topology

I am working on a problem in my advanced algebra class and am having 
trouble finding information on the subject. My local library does not 
have many math references.

Here is the problem.

A. Show that a triangle, a square, or any polygon is homeomorphic to 
   a circle.

B. The function f(x) = tan(x) is a homeomorphism if the domain of 
   the function is the open interval (-PI/2, PI/2). What two sets of 
   points does this homeomorphism show to be topologically equivalent?

C. The term sterographic projection is used to refer to a particular 
   correspondence between the points that make up a sphere minus the 
   north pole and the points that make up an infinat plane. Describe a 
   stereographic projection projection and justify why it is a 
   homeomorphism.

I have no texts that describe homeomorphism or stereographic 
projection.  Any help would be appreciated.

Thanks,
Kristin


Date: 12/31/97 at 1:20:05
From: Doctor Jerry
Subject: Re: Topology

Hi Kristin,

I'll make some comments.

>A. Show that a triangle, a square, or any polygon is homeomorphic to 
>   a circle.

Two spaces X and  Y are homeomorphic if there is a 1-1 function f 
from X onto Y such that f and f^{-1} are continuous. The function f 
is called a homeomorphism. The above problem can be reduced to 
showing that the unit circle is homemorphic to the circumscribed (or 
inscribed) n-gon. This can be done with a central projection, so that 
a point P on the circle corresponds to the point Q on the polygon 
which is the intersection of the line through P and the center of the 
circle.

>B. The function f(x) = tan(x) is a homeomorphism if the domain of 
>   the function is the open interval (-PI/2, PI/2).  What two sets of 
>   points does this homeomorphism show to be topologically  
>   equivalent.

X = (-pi/2,pi/2) and Y = R.  f is continuous, 1-1, and onto; moreover, 
arctan is continuous.

>C. The term stereographic projection is used to refer to a particular 
>   correspondence between the points that make up a sphere minus the 
>   north pole and the points that make up an infinat plane. Describe  
>   a stereographic projection projection and justify why it is a 
>   homeomorphism.

The stereographic projection can be described as follows: let P be 
any point on the sphere minus the north pole; draw a line from the 
north pole through P and piercing the plane at Q. Then f(P)=Q. It's 
not hard to work out a formula for f. Just figure it out in, say, the 
(y,z)-plane, using spherical coordinates for the sphere. The function 
should be independent of theta and depend only upon phi.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 12/31/97 at 09:03:08
From: kristin bullard
Subject: Re: Topology

Is there a simple definition for homeoporphism? I can't seem to find 
one. Is there a simple definition for topology?  

Thanks,
Kristin


Date: 12/31/97 at 17:06:09
From: Doctor Jerry
Subject: Re: topology

Hi Kristin,

No doubt I didn't explain well enough. I think I saw the word topology 
and irrationally assumed that you were taking a topology course.  
Sorry. I'll try to give a simpler explanation of homeomorphism and of 
topology.

Two geometric objects are homeomorphic when they have the same 
structure. Now, what does it mean to say that they have the same 
structure?  Earlier, I gave the a rigorous definition. More loosely, 
two objects have the same structure if either can be deformed into the 
other "continuously," that is, without tearing. Also not allowed is 
two or more points of one of the objects being deformed onto one point 
of the other.

A solid doughnut is like a solid coffee cup with handle. You can 
visualize how either of these could be deformed into the other without 
tearing or duplication.

The letters b and p are homemorphic, but the letters o and l are not. 

A circle and a polygon are homemorphic since either can be 
continuously deformed onto the other. I partly gave the idea of the 
deformation: shrink or expand the circle until it is the unit circle.  
Shrink or expand, lengthening or shortening legs of the polygon until 
it is a regular polygon circumscribed about the unit circle. Then, the 
continuous deformation of the polygon can take place along rays from 
the center.

An object can have different "topologies" placed on it. The most 
natural topology for a straight line or a circle comes from the 
ordinary euclidean distance between the points of the line or the 
points of the circle. One can tell how far or near one point is from 
another with these distance functions. If the object is a rectangular 
grid of city streets, then the ordinary distance function can again be 
used. Also used is the "taxicab" distance function. For this, the 
distance from one point to another is the shortest possible distance 
(measured by odometer) between the points as measured by a taxicab 
driving on the streets. This leads to a somewhat different look to the 
object. 

The Encylopaedia Brittanica often has good material on mathematical 
topics. You  might try looking up topology. There are several branches 
of topology. Your questions occur within what are called "metric 
spaces,"  in which one is given some kind of distance function.  

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   


Date: 01/04/98 at 21:46:23
From: kristin bullard
Subject: Re: Topology

I have another term that I need to have explained in simple terms. 
What is stereographic projection? And why would it be a homeomorphism?

Kristin


Date: 01/05/98 at 08:20:45
From: Doctor Jerry
Subject: Re: Topology

Hi Kristin,

In three space, we'll consider a sphere of radius 1 and resting upon 
the (x,y)-plane at (0,0,0). The stereographic projection of this 
sphere (with its north pole removed; I'll call the sphere with north 
pole removed the punctured sphere) onto the (x,y)-plane is just this: 
Choose any point P on the punctured sphere. Draw a line from the north 
pole through P. Let the point on the (x,y)-plane through which this 
line passes be Q = f(P). As P varies over the puncatured sphere, Q 
will vary over the entire plane. The function f is a homeomorphism.  
It is a one-to-one continuous function mapping the punctured sphere 
onto the plane; hence these two spaces are homemorphic. I can describe 
the homemorphism in spherical coordinates.

In spherical coordinates, the sphere can be described by the equation 
(I'll use r for rho, p for phi, and t for theta, where (r,p,t) are 
the spherical coordinates of a point) r=2*cos(p). Let P=(r,p,pi/2) 
be any point on the trace in the (y,z)-plane of the punctured sphere. 
Then r = 2*cos(p). Draw a triangle with vertices the north pole, P, 
and Q. A little trig shows that the spherical coords of Q are 
(2cot(p),pi/2,p/2). From the symmetry of the sphere about the z-axis, 
we can give the general form of f. For any point (r,p,t) on the 
punctured sphere,

   f(r,p,t) = (2cot(p),pi/2,t).

Given that 0<p< = pi/2 on the punctured sphere, f is continuous.  
It can be shown to be one-to-one. 

So, the punctured sphere and the plane have the same structure (but 
not the same shape.

The stereographic projection is used for some mapmaking.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
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High School Higher-Dimensional Geometry

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