- p.7 #2 -- "get-acquainted special"
- p.8 #5
- p.8 #9
- p.12 #3
- p.12 #5
BONUS p.13 #7 (as corrected)
- p.15 #4 (omit "disjoint")
- p.15 #8
- p.18 #5
- p.25 #3
- p.25 #7
- p.27 #1
- p.28 #7 first part only.
BONUS p.28 #11.
- p.36 #5 first part only; extra credit for second part.
- p.38 #13
BONUS Is a bounded linear operator also characterized as mapping bounded
sets to bounded sets? (Proof or counterexample.)
- p.45 #1
- For a metric space (X,d), prove the Hausdorff distance
h is a metric on the hyperspace of compact subapaces of
X. For Extra Credit, prove that h is complete if
d is.
- Prove the union property of the Hausdorff metric as given in class.
BONUS Do we have enough tools to prove the density of the dyadic fractions
in [0,1]? For example, does it follow from the example in class?
- Prove directly that the Cantor set is indeed the attractor of the
example presented in class.
- Prove that if X is compact, so is the hyperspace of compact subspaces.
- Do the problem about the triangular seed for the extension of the
Cantor set contraction to the real plane.
- Prove that a decreasing sequence of nested compact sets in
X converges in the hyperspace. Determine when the limit equals
the intersection; if not always, what else can happen?
- Koch's Curve, all three parts.
- Let X be compact. Prove that an increasing sequence of
nested compact sets is Cauchy, hence converges in the hyperspace.
Let K be a compact set that is contained in the image of
every compact set A under some contraction Gamma. Express the
attractor of Gamma using the union of all the iterates of K
under Gamma.
BONUS If X is not compact, precisely when does such a
sequence converge, and how does the limit relate to the union?
- Do the problem about the circular seed.
- Cantor Function #1
- " #2
- " #3
BONUS How many topologies are there on a finite set of n points?
- Cantor Function #4 -- due Mon 18 May by 4:30 pm
- Find two topological spaces X and Y such that
every function from X to Y is continuous, while
the only continuous functions from Y to X are the
constants. -- due Mon 18 May by 4:30 pm
- Prove that Sierpinski space S is path-connected. -- due Mon
18 May by 4:30 pm