## MAT530 Final Examination - 3 hours - December 13, 1995

* Each part of each question is worth 10 points. Total = 110.*

** 1** Give an example of a continuous, injective map
*f*: R --> R^2 which
is not a homeomorphism onto its image.
** 2** Prove that in a normal space, given two disjoint
closed sets *A*,*B*, there exist disjoint, closed sets *A*',
*B*'
with *A* contained in the interior of *A*' and *B*
contained in the interior of *B*'.

** 3** ** a** Prove that if *A*,*B*
are disjoint, compact sets in R^2,
then (*) there exist points *a* in *A* and *b* in *B*
such that
d(*a*,*b*) < = d(*a*',*b*')
for any other such pair of points. (d is the Euclidean distance).

** b** Same problem, but now suppose *A*,*B* are closed, and only
*B* is known to be compact. Prove (*).

** c** Show by an example that (*) fails to hold under the
weaker hypothesis that *A*,*B* are closed.

** 4** Consider the set of *all*
functions *f*: R --> [0,1]. Since a function is
determined by its values at every point, this set may be identified
with [0,1] times [0,1] times ..., one factor for each *t* in R.
Give this set the product topology, and call the resulting space
* F*.

** a** Give an example (not the whole space!) of a neighborhood
of the function sin^2 *t* in *F*.

** b** Consider the set *B* contained in *F* of functions
which are everywhere < = 1/2. Show that *B* is closed
in the product topology.

** 5** ** a** Suppose *X* is an arc-connected space, and
that *p*: *Y* --> *X* is a covering map. Suppose
one point *x* in *X* has *k* pre-images. Prove that every point
of *X* has *k* pre-images. (Such a map is called a *k*-fold
cover).

** b** Exhibit two non-homeomorphic, *connected* 3-fold covers of the
Figure-8 space. Prove carefully that each is a covering
and that they are not homeomorphic.

** 6** In R^3 with coordinates *x*,*y*,*z*, let *S* be
the unit circle about the origin in the (*x*,*y*)-plane.

** a** Prove that R^3 - {*z*-axis} has *S* as
deformation retract.

** b** Prove or disprove: R^3 - *S* has the *z*-axis as
deformation retract.