Conference rusure::math

>    Here is a puzzle I think is interesting.  To those who knows a lot
>    about manifold theory or Algebraic Topology, please do not enter
>    the solution right away.
>    
>    Construct a topological space that is locally homeomorphic to R^n
>    (you can choose the n) but is not Hausdorff.

	You might increase the mumber of people who can have a shot at this
question if you briefly define:

	(1) topological space
	(2) locally homeomorphic
	(3) Hausdorff

Thanx,
Andrew Buchanan

>	You might increase the mumber of people who can have a shot at this
>question if you briefly define:
>
>	(1) topological space

        Definition see note TOPOLOGY
    
>    	(2) locally Homeomorphic
    
    Two spaces are homeomorphic to each other if they are topologically
    identical.  Locally homeomorphic means that for each point in
    the space there is an open neighbor that is homeomorphic to another
    space (in our case, it is R^n).

>    (3) Hausdorff
 
    A topological space is Hausdorff if for any two points in the space
    there exist disjoint open sets such that each open set contains
    only one of the above chosen point.
    
    Well here are some brief definitions.  I know I am not doing a good
    job explaining these terms.  Will any one on the net give it a try?
    Eugene   

     If (X,T1) and (Y,T2) are topological spaces, and f:X -> Y
     is a function from X to Y, then f is said to be continuous
     if for every open subset V of Y, its inverse image under
     f is open in X.
     
     [Note that the "epsilon-delta" definition of continuity
     for functions from the reals to the reals agrees with
     this definition.]
     
     The function f is a homeomorphism if it is a bijection
     (in older terminology, one-to-one and onto) and both
     f and its inverse function are continuous.
     
     The spaces X (the usual abbreviation for the ordered pair
     (X,T1)) and Y are homeomorphic if there is a function f:X ->
     Y which is a homeomorphism between them.  Essentially f
     shows how to associate to each element of X an element of Y
     (and vice versa) such that the two topologies are
     essentially identical, like renamings of each other. 
     
     X is locally homeomorphic to Y if each point of X is
     contained in an open set U of X which [when given the
     subspace topology] is homeomorphic to Y.
     
     As an example, the set of all reals with the Euclidean
     topology is homeomorphic to the set of reals (0,1) =
     { x | 0 < x < 1 } with the Euclidean topology.  Or the
     surface of a sphere but with one point missing is
     homeomorphic with the Euclidean plane.
     
     Dan

    Ok, I think people have had adequate time to think about the problem,
    so for those of you who have solved it, you are welcome to enter
    your solution(s).
    Eugene

     An example with n = 1.
     
     Let X be the space of all nonzero real numbers, plus
     two extra elements called 0-top and 0-bottom.
     
                                    0-top
           ------------------------) (-------------------------
                                    0-bottom
     
     The topology is given by the following basis of open
     sets:
     
          For a,b nonzero reals that are both positive or
          both negative, (a,b) = {x | a < x < b} is open.
     
          For a < 0 and b > 0, both
               {x | a < x < 0} U {x | 0 < x < b} U {0-top}
               {x | a < x < 0} U {x | 0 < x < b} U {0-bottom}
          are open.
     
     Arbitrary unions of basis elements are open.
     
     Then X with this topology is locally homeomorphic to the
     real line, but is not Hausdorff because 0-top and 0-bottom
     cannot be separated by open sets.  Essentially this space is
     a real line with two zeroes.  It is usually given as the
     real line plus one extra point, 0-imposter, but as the two
     0's are topologically equivalent that doesn't seem fair.
     
     Dan 

    Re .5
         Good job.  Another way of looking at it is to think of it as
    the quotient topology of two real lines identified with each other
    except at 0.  Incidentally, if you want to add the constrain of
    the manifold being compact, then you use circles rather than the
    real lines.  
    Eugene