Topology Problem Sets

Problem Set 1

This problem set will be due Friday, February 3.  I apologise for the delay.  I've tried to keep it simple as a consequence.

§3.2 #4

Create a topological space with at least three elements that is not T1/2, and create one that is T4 (normal).

Choose one from §3.4 7, 8, 9

Problem Set 2

This problem set will be due Friday, February 17. 

§3.5 1
§3.6 4
§3.7 6

Define an equivalence relation on the plane R2 as follows:  (x1, y1) ~ (x2, y2) if x1+ y12 = x2+ y22.  Let R2* be the identification space resulting from the quotient topology.  This is homeomorphic to a familiar space; what is it?   Repeat the previous question for the relation given by x12+ y12 = x22+ y22.

Problem Set 3

This problem set will be due Friday, March 3.

Choose one from:  §4.2 2, 5 (you may use either definition of boundary without proof)

We have proven if f:[0, 1] -> [0, 1] is continuous, then there is a fixed point f(x) = x.  Is this still true for f: (0, 1] -> f: (0, 1]?  Discuss and explain.

§4.5 1

Choose one from:  §4.6 1, 4

Problem Set 4

This problem set will be due Friday, March 24.

§5.2 6 (recall 'identification' is identical to a quotient map)
§5.3 2

A topological space X is said to be locally compact if each point x in X has a compact set N containing an open set U containing x for each x  [thus x in open U in compact N].  Prove that the real line and Rn are locally compact.  (This is restated §5.4 2 FYI, a neighbourhood is a set containing an open set containing the point.)

Let X be a locally compact Hausdorff space.  Take some object outside X, call it "infinity" denoted by a "lazy 8" (I can't typeset that on-line, but you know the infinity symbol).  Consider Y = X union {infinity}.  Create a topology on Y by defining the collection of open sets in Y to be all sets of the following types:
  (1)  U, where U is an open subset of X,
  (2)  Y \ C, where C is a compact subset of X.
Y is called the one-point compactification of X.
  1. Prove that this in fact defines a topology on Y.
  2. Prove that Y is compact.
  3. Prove that the one-point compactification of R is homeomorphic to S1.
(This is a restatement of §5.4 3)

Surface Problem Set

This will be due Friday, April 7.

§5.7 4 (finish 4 by showing that K = P # P), 5, 6

Make a model of a Möbius strip.  Cut along a line one-third from the boundary.  Describe the results.

Identify the surface represented by this code:  CBB-1ADA-1C-1D-1

Identify the surface represented by this code:  EECBB-1ADA-1C-1D-1 (ahh, that changes everything!)

Fundamental Group Problem Set

All questions from Goodman handout:  6.1.4, 6.1.20, 6.2.1, 6.2.2, 6.3.5