Homework 13, due May 3
1. (based on 36.2) Let X be a compact Hausdorff space. Suppose that for each x in X, there is a neighborhood U_x of x that can be embedded into some Euclidean space (of finite dimension). Prove that all of X can be embedded into Euclidean space.
2. (38.3) Under what conditions does a metrizable space have a metrizable compactification?
3. (38.4) Let Y be an arbitrary compactification of X. Let beta(X) be the Stone-Chech compactification. Show that there is a continuous surjective closed map (in particular a quotient map) g:beta(X)->Y that equals the identity on X.
4. (38.8) Show that beta(Z_+) has cardinality at least I^I, where I=[0,1]. (Hint: I^I has a countable dense subset.)

Homework 12, due April 26
1. (34.3) Let X be a compact Hausdorff space. Show that X is metrizable if and only if it has a countable basis.
2. (35.1) Show that the Tietze extension theorem implies the Urysohn Lemma. (That is, assume you have proven the Tietze extension theorem, and use it as a shortcut in proving the Urysohn Lemma.)
3. (35.2) In the proof of the Tietze extension theorem, we subdivide the interval [-r, r] into I1=[-r,-ar], I2-[-ar,ar], I3=[ar,r], using a=1/3. Which values could we have used? Classify all numbers in (0,1) as usable/unusable.
4. (35.3) Let X be metrizable. Prove that the following are equivalent: (i) X is bounded under every metric that gives he topology of X, (ii) Every continuous function to R is bounded, (iii) X is limit point compact. (Hint: If f: X->R is a continuous function, then F(x)=(x,f(x)) is an imbedding of X in XxR. If A is an infinite subset of X having no limit point, let f be a surjection of A onto Z_+.)

Homework 11, due April 19
1. (32.3) Show that every locally compact Hausdorff space is regular.
2. (32.4) Show that every regular Lindelof space is normal.
3. (33.3) Give a direct proof of the Urysohn lemma for a metric space (X,d) by setting f(X)=d(x,A)/(d(x,A)+d(x,B)).
4. (33.4) Recall that a G-delta set is the intersection of a countable collection of open sets. Suppose X is normal. Prove that there exists a continuous function f: X->[0,1] such that f(x)=0 for x in A, and f(x)>0 if x is not in A, if and only if A is a closed G-delta set in X.
5. (33.6a) Show that every metrizable space is perfectly normal: it is normal and every closed set in X is G-delta.

Homework 10, due April 12
1. (29.1) Show that the rationals Q are not locally compact.
2. (29.6) Show that the one-point compactification of R is homeomorphic to S^1.
3. (30.13) Show that if X is separable, then every collection of disjoint open sets in X is countable.
4. (30.14) Show that if X is Lindelof and Y is compact, then XxY is Lindelof.
5. (31.1) Show that if X is regular, then every pair of points of X have neighborhoods whose closures are disjoint.

Homework 9, due April 5
1. (27.2a) Show that [0,1] is not compact in the K-topology (recall that K={1/n : n in N}).
2. (based on 28.2) Prove that [0,1] not limit point compact in R_l (the lower-limit topology). Is it compact?
3. Use Theorem 26.6 to prove that R^2/Z^2 is homeomorphic to your favorite torus.
4. Let X be a metric space, and Y the collection of all compact subsets of X. Given two points a,b in Y (that is, compact subsets of X), define the Hausdorff distance d(a,b) as in Homework 4. Prove:
a) For each a,b in Y, d(a,b) is finite.
b) For each a,b in Y, d(a,b)=0 if and only if a=b.
c) d satisfies the triangle inequality.
d) If X is R, prove that the intervals [0,1+1/i] converge to the interval [0,1] in the Hausdorff metric.
e) Prove that diameter is a continuous function on Y.

Homework 8, due March 29
1. (Based on 26.7) Prove that if Y is compact, then the projection mapping from XxY to X is closed, for any space X. (Hint: tube lemma.)
2. (Based on 26.8) Let f: X->Y be a function from some space to a compact Hausdorff space. Prove that f is continuous if and only if the graph of f is closed. (Hint: use the previous result.)
3. List all the discrete and compact spaces, up to homeomorphism.
4. Prove that any set with the finite-complement topology is compact.
5. (Based on 26.11) Let X be a compact Hausdorff space, and A_i a nested sequence of closed connected subsets. Prove that their intersection is connected.
6. Read Theorem 27.7. Use it to prove that the Wallis sieve is uncountable, and the previous problem to prove that it is connected. (Extra nerd points: find a way to do this by generalizing Cantor's argument, using an extra page.)

Homework 7, due March 22
(Note: due on Thursday by request.)
1. Consider the equivalence relation on R^2 given by setting (x,y) equivalent to (x', y') if (x-x', y-y') is in Z^2.
a. Describe the corresponding quotient space X (hint: it's enough to think about the unit square).
b. Find an explicit homeomorphism between X and a surface in R^3 (hint: use lots of trig functions).
2. (24.9) Assume R is uncountable. Show that if A is a countable subset of R^2, then its complement is path-connected. (Hint: how many lines are there passing through a given point in R^2?)
3. (Example of a "character variety") Consider the space SL(2,R) of 2-by-2 real matrices with determinant 1. Define an equivalence relation on SL(2,R) by considering A and B to be equivalent if they are conjugate via some matrix C in GL(2,R) (in other words, A and B are the same up to a change of coordinates). Let X be the associated quotient space.
a) Topologize SL(2,R) using an embedding in R^4.
b) Prove that the trace function induces a continuous function from X to R.
c) Classify points of X based on their trace into three types: rotations, shears, and hyperbolic transformations (try some examples first). Explain how each one behaves.
d) It turns out that the trace function is ALMOST a homeomorphism between X and R: one fiber has two points. Find two distinct elements of X that have the same trace.
e) Prove that X is not Hausdorff. (Hint: conjugate a non-identity matrix to make it closer to the identity matrix).

Homework 6, due March 6
1. (23.4) Show that if X is an infinite set with the finite-complement topology, then it is connected.
2. (23.5) A space is "totally disconnected" if its only connected subspaces are one-point sets. Show that if X has the discrete topology, then X is totally disconnected. Is the converse true?
3. (based on 23.11) A "fiber" of a map p:X->Y is the preimage of a point in Y. Suppose that p:X->Y is a quotient map.
a) Prove that if X is connected, then Y is connected.
b) Prove that if Y is connected and every fiber of p is connected, then X is connected.
c) Prove that the assumption about the fiber in (b) is necessary.
4. (based on 24.1)
a) Prove that no two of the spaces (0,1), (0,1], and [0,1] are homeomorphic. (Hint: remove a point or two and see what happens.)
b) Suppose X embeds into Y and vice versa. Are X and Y homeomorphic?
c) Show that R^n and R are not homeomorphic if n>1.
5. (24.2) Let f: S^1 -> R be a continuous map. Show that there exists a point x in S^1 such that f(x)=f(-x). Hint: use a slightly modified version of f, and connectedness.

Homework 5, due March 1
(Note: due on Thursday because of limited office hour availability.)

1. Prove that the box metric on R^n is a metric. Make sure that the proof still works if each copy of R is replaced with a different metric space, and make a comment confirming this.
2. Two metric spaces are considered the same if there is an isometry between them. Come up with reasonable definitions for "isometry" and "isometric embedding". Provide an example of two metric spaces that are homeomorphic but not isometric.
3. (Simplified Hausdorff metric) Consider the set X of closed segments in the Euclidean plane R^2 (which is equipped with the usual Euclidean metric).
a) Given a segment x in X, let N(x,r) be the set of points that are distance at most r from x. Describe the shape of N(x,r).
b) Given two segments x and y, prove that there is an r such that x is contained in N(y,r).
c) Given two segments x and y, let d(x,y) be the smallest r such that x is contained in N(y,r) and y is contained in N(x,r). Prove that d is a metric on X.
d) The Hausdorff metric can be defined, in the same way, on the space Y of all closed and bounded subsets of R^2 (do not prove this). Explain why it can't be defined on the space Z of all closed subsets of R^2 (hint: look close to infinity). Explain also why it can't be defined on the space W of bounded open subsets of R^2 (hint: puncture a disk).
4. Prove that an uncountable product of metric spaces is not metrizable, as long as each one contains at least two points.

Homework 4, due February 22
(Note: due on Thursday because the problems were posted late.)

1. It seems plausible that a function is continuous if and only if its graph is closed. Disprove both directions.
2. (Exercise 18.8) Suppose f and g are continuous functions from some space X to an ordered space Y.
a) Prove that the set of points in X where f is smaller or equal to g is closed in X.
b) Define a new function h by taking the minimum of f and g. Prove that h is continuous.
3. (based on Exercise 18.13) Suppose A is a subset of X, and f is a continous function from A to a Hausdorff space Y.
a) Provide an example in which f does not extend continuously to the closure of A.
b) Prove that if f does extend to the closure of A, then it extends in a unique way. (Hint: it's enough to think about one point in the boundary of A.)
4. (Theorem 19.4) Prove that an arbitrary product of Hausdorff spaces is Hausdorff, for both the box and product topologies.

Homework 3, due February 13

1. A function is "open" if the image of every open set is open. A function is "closed" is the image of every closed set is closed. Prove that the projection mappings from XxY to each coordinate are continuous and open but generally not closed. Are all continuous functions open?
2. Let X be an ordered set. If Y is a proper subset of X that is convex in X, does it follow that Y is an interval or a ray in X?
3. Show that the dictionary order topology on R^2 is the same as the product topology on (R_d)xR, where R_d is the real line with the discrete topology.
4. Let {A_i} be a collection of subsets of some topological space, and let A be their union. Prove that the closure of A contains the union of the closures of each A_i. Give an example where equality fails.
5. Show that a space X is Hausdorff if and only if the diagonal in XxX is closed.

Homework 2, due February 6

1. Consider the real numbers with the order topology. Prove that every open set is a countable union of open intervals. (Hint: find a countable basis.) Is it true that the complement of every open set is a countable union of closed intervals? (Hint: the Cantor set contains no intervals.)
2. (Based on Exercise 13.4) We showed that that the finite-complement topology is in fact a topology, and for infinite sets is neither discrete nor indiscrete. Make reasonable definitions for countable-complement and infinite-complement topologies. Discuss different ways in which each of these could possibly be defined, and how you chose your definitions. Prove that, with your definitions, each of these is a topology. Give interesting examples of sets with these topologies.

Homework 1, due January 30

1. Prove that every vector space has a basis.
2. Why are you here? Tell me a bit about how this class fits into your career path. (Note: Don't write a poem or a biography, and include some math.)
Note: for both questions, think of your classmates as the audience. Include all necessary details, but keep the response as concise as possible.