338 Problem Sets

As the problem sets are created they will be posted here. 

Problem Set 1  is due 9 September. 

1. What is the minimum number of possible open sets in a topology T?  Give an example of a set with this topology.  Create examples of topologies with n open sets, for each n ≥ the minimum you found above. 


2. Prove that the set of closed intervals [a,b] satisfies the necessary conditions for being a basis of the real line.  How does the topology on the real line generated by the basis consisting of all sets of the form [a,b] for real a,b compare to that generated by the basis consisting of all sets of the form (a,b)?  [The latter is the standard topology for the real line.  In particular, what topology does [a,b] generate (hint: it has a name)?

3. Prove TB =TB′ if and only if for every x∈X, if x∈U∈B, then there exists a V ∈B′ such that x∈V ⊂U, and for every x∈X, if x∈V ∈B′,then there exists a U ∈ B such that x ∈ U ⊂ V.


Problem Set 2 is due 23 September.

0. Find examples of topological spaces with at least three elements that demonstrate the separation axioms are distinct. That is, find an example that satisfies none of them, an example that is only T0, an example that is T0 and not T1, and so forth.  This problem gets quite difficult.  Take it as far as you dare.

1. M3.4.13

2. Consider the real numbers with the finite complement topology (example 6 on page 72 of M), and N the natural numbers, and B = {1,2}.  Find the closure, interior, and boundary for both N and B in this topology. 

3. Prove that if f : X → Y is continuous and if S is a subspace of X, then the restriction f|S : S → Y is continuous.

4. Give an example of sets B⊂A⊂R3 where B is open relative to A but not open in R3.

5. Prove or disprove the continuous image of a Hausdorff space still Hausdorff.  Suppose X = X1 × · · · × Xn, where each Xj is nonempty. Prove that if X is Hausdorff, then each Xj is Hausdorff.

6. The Pasting Lemma Let X be a topological space with closed subsets A and B such that X = A∪B. Let f : A → Y and g : B → Y be continuous functions such that for each x∈A∩B, f(x)=g(x).  Define a new function f ∪g:X→Y by

f∪g(x)= {f(x) for x∈A, g(x) for x∈B

(1) Prove that f ∪ g is continuous.
(2) Give an example to show that the condition that A and B must be closed is necessary.
(3) Why is this called the pasting lemma?

(this is here as continuity 7 if you don't like the typesetting)




Problem Set 3 is due 30 September.

0.  Either:

Let ∼ be an equivalence relation on a topological space X. Prove that X/∼ is a T1-space if and only if each equivalence class is closed. Give an example of a T1-space X and an equivalence relation ∼ such that X/∼ is not a T1-space.

or
 
For n ≥ 1, define Pn = Sn/ ∼, where the equivalence relation is defined by declaring x∼y if and only if x=y or x=−y. In other words, Pn  is obtained from Sn  by identifying pairs of antipodal points. The space Pn is called real projective space of dimension n, and it can be regarded as the set of lines in Rn+1 which pass through the origin. Establish the following assertions:

(a) Pn is a Hausdorff space.
(b) The projection π : Sn → Pn is a local homeomorphism, that is each x ∈ Sn is contained in an open set that is mapped homeomorphically by π onto an open set containing π(x).
(c) P1 is homeomorphic to the circle S1.
(d) Pn is homeomorphic to the quotient space obtained from the closed unit ball Bn in Rn by identifying antipodal points of its boundary Sn−1.

1. CS1.3

2. CS1.4

Problem Set 4 is due 17 October.

CS2.3, CS3.1

3. Consider a wire frame for a cube.  Encircle the wire with a small round tube (imagine a pool toy made in the shape of the frame of a cube).  This is an orientable surface.  As such it is homeomorphic to a connected sum of tori.  How many?  Justify with drawings.

4. In the following model of a projective plane, what is the result of cutting along the closed curve alpha?  what is the result of cutting along the closed curve beta?  what is the result of cutting along both closed curves?  Cut Projective Plane

CS3.4

CS4.5:  What compact surfaces have non-negative Euler characteristic?  Give a complete list.  Show that any cell decomposition on a compact surface of negative Euler characteristic must contain either a face with at least five edges or a vertex of valence at least five.  

7. In our classification of surfaces proof, we first proved that if M is a word for an orientable compact surface and the length of M is 2n, then M ~ mT for some m ≥ 0 satisfying 4m ≤ 2n.  Then we proved if M is a world for a nonorientable compact surface and the length of M is 2n, then M is homeomorphic to a word of the form mP for some m ≥ 1.  I noticed that we have dropped the "satisfying" clause at the end the second time.  Bring it back.  Find the correct bound on m and prove it.


Problem Set 5 is due 24 October.

1.  M4.2.5

2.   Give examples of sets A and B in R2 which satisfy:

(a) A and B are connected, but A ∩ B is not connected.
(b) A and B are connected, but A\B is not connected.
(c) Neither A nor B are connected, but A ∪ B is connected. 

3.  Let X/∼ be the quotient space determined by an equivalence relation ∼ on a topological space X. Prove that if X is connected, then X/∼ is connected. 

Problem Set 6 is due 31 October.

(costumes accepted)

1.  M4.2.2 and 4.2.3 as one problem. 

2. M4.6.3 or  Suppose X = X1 × · · · × Xn, where each Xj is nonempty. Prove X is path-connected if and only if each Xj is path-connected. 

3. Prove or disprove:

(a) The intersection of a collection of compact subsets of a space is compact.
(b) The intersection of a collection of compact subsets of a Hausdorff space is compact.

Problem Set 7 is due 7 November.

0. Replacement problem - I was wrong - local compactness is *not* inherited for subspaces.  Find as many properties as you can that we have discussed that are.  Find some that are not - you don't need to prove either.  Build a subspace of the reals with the standard topology that is not locally compact.  Prove that it is not.

1-2. (scored as two problems) Let X be a locally compact Hausdorff space. Take some object outside X, call it ∞. Consider Y = X ∪ {∞}. 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. Prove that this in fact defines a topology on Y . Prove that Y is compact. Prove that the one-point compactification of R is homeomorphic to S1. (This is a restatement of §5.4 3)  What do you get when you take the one point compactification of something already compact?  Try [0,1]. 

Problem Set 8 is due 14 November.

0. The Poincare dodecahedral space is a famous three manifold obtained from a dodecahedron by gluing opposite faces.  Because they don't align on opposite faces, there must be some transforming.  We will glue them with as little twist as possible - 1/10 a turn in the clockwise direction from front to back.  (Make sure you do the same thing for all faces.)  This attaches the faces in pairs.  How many edges are there originally on the dodecahedron?  How many edges come together in each set?  How many vertices come together in each set?  Are the points locally homeomorphic to R^3? 

1. M4.7.3

2. M4.7.4

Problem Set 9 is due 21 November.

2. Generous problem.  Give an example of a subset A of a topological space X such that A is a retract of X, but their fundamental groups are not isomorphic.  Justify details (feel free to use our intuitive fundamental group computations if you need them).

3. M4.7.5

4. M4.7.6

5. G5.1.3

Problem Set 10 is due 5 December.

This is your vector field problem set. 

1. G5.2.1

2. G5.4.7 or G5.4.9

3. G5.5.9

4. G5.5.20 or G5.5.21