Midterm Exam I
Part II: TAKE HOME
MATHS 441
Dr. Fischer
Due: September 25, 2009
6. Let X, Y ⊆Rn. When you ask four different students as to when a function
f:X→Yshould be called continuous, you might get four seemingly different
answers. For example:
(i) Audra might say: “fis continuous if for every x∈Xand every > 0 there
is a δ > 0 such that d(f(x), f (z)) < whenever z∈Xand d(x, z)< δ”;
(ii) Cameron might say: “fis continuous if for every x∈Xand every sequence
(xk) in Xwhich converges to x, the sequence (f(xk)) converges to f(x)”;
(iii) David might say: “fis continuous if for every subset C⊆Ywhich is
closed in Y, the set {x∈X|f(x)∈C}is a closed in X”;
(iv) Jie might say: “fis continuous if for every subset U⊆Ywhich is open
in Y, the set {x∈X|f(x)∈U}is open in X”.
Prove that all of the above statements are in fact equivalent.
7. Let X⊆Rn.
Prove that Xis compact if and only if Xis both closed in Rnand bounded.
8. Suppose Xand Yare two compact surfaces. Prove that X#Yis independent
of the direction along which one glues the two boundary circles together.
9. Consider the following compact surfaces, each represented by a plane model.
X:a b−1cda−1e f−1g−1e−1h−1ibd−1c−1i−1f g h
Y:e d e f a−1b c−1d−1f a c−1b
Z:bec−1d−1f g−1e−1d−1a−1b f g−1c−1a−1
Find the normal form and the alternative normal form for each of these surfaces.
10. Consider the following cut-and-paste move on a plane model for a surface X,
which has 4 or more edges: “Focus on two adjacent edges, say aand b, which
are not being identified. Cut off a triangle, two of whose edges are aand b, and
the third of which is your cut x. Locate the other edge labeled a±1and glue the
triangle back along the two a-edges.” Based on this procedure, derive two rules:
(i) If Xis represented by abAaB, then Xis also represented by ...
(ii) If Xis represented by abAa−1B, then Xis also represented by ...
Show that, using only “cancellations” and moves of the above type, one can
either transform the plane model into one where all vertices are identified to
one point, or else Xis homeomorphic to S2.
[Hints are attached!]
