Note: below I use "^" to indicate superscripts and "_" to indicate subscripts! Will change soon...
1: Use the "open set" definition of continuity to check that the composition gf of continuous functions f: X -> Y, g: Y -> Z (between topological spaces X,Y,Z) is continuous.
2: Construct homeomorphisms between the following topological spaces (their topologies are defined using their natural metrics as subspaces of Euclidean spaces):
(i). (0,1) and R
(ii). S^1 - {(1,0)} and S^1 - {(0,1)}
(iii). S^2 - {(0,0,1)} and R^2
3: Let f: R -> R be a continuous function, and consider its graph G = {(x, f(x)): for all x in R} inside R^2. Show that G is homeomorphic to R.
4: Let X={a,b,c} be a set with three elements. How many ways are there to define a topology on X? How many different homeomorphism classes do these fall into?
5: Classify (intuitively - no proof required) the capital letters of the alphabet into homeomorphism classes. (The answer will depend on exactly how you write your letters, so be careful!)
Homework 2, Jan 26; due Feb 2
1: Suppose X is a topological space, and A is a subset, which we can also regard as a space in its own right using the subspace topology. Let M be a subset of A.
(i). Show that if A is closed in X, and M is closed in A, then M is closed in X.
(ii). Show that if A is open in X, and M is open in A, then M is open in X.
(iii). Give examples to show that if A is open in X and M closed in A (or vice versa) then M doesn't have to be open or closed in X.
2: Suppose P, Q are topological spaces and A, B are subsets of P, Q respectively. The subset A X B of P X Q can be made into a topological space in two ways: as a product of subspaces, or as a subspace of the product. Prove that these are homeomorphic.
3: Let D be the closed unit disc in R^2, and let ~ be the equivalence relation which identifies together all points in the boundary circle of D. Prove that D/~ is homeomorphic to the unit sphere S^2.
4: Prove the "gluing lemma": if a space X is written as a union of finitely-many closed sets Fi, and fi:Fi -> Y are functions to another space Y which are continuous (with respect to the subspace topologies of the Fi) and agree where Fi overlap, then the union of the fi defines a continuous function f:X -> Y.
5. (i). Recall that the closure of a subset A in a space X is the set of points in X, all of whose neighbourhoods meet A. Prove that this is the same as the intersection of all closed sets containing A, and hence that the closure of a set is a closed set!
(ii). Similarly, one can define the interior of a subset A as the set of all points x in X such that A is a neighbourhood of x. Prove that this equals the union of all open subsets contained in A, and hence that the interior of any set is open.
(iii). (More challenging). Consider the operations C=closure, I=interior, N=complement (N is for negation!) as functions which take subsets of (the real numbers) R to subsets of R. Find a subset A such that CI(A) and IC(A) are distinct, showing that the semigroup generated by these operations is not commutative. Prove that in this semigroup C2 = C, I2 = I, N2 = identity. Can you find any other relations? Prove that the semigroup is finite, and find its order!
Homework 3, Feb 2; due Feb 9
1: Show that the union of finitely-many compact subsets of a space is compact.
2: Show that an infinite subset of a compact space must have an accumulation point.
3: Show that any continuous map from a compact space to a metric space is bounded.
4: Suppose K is a compact subset of a metric space X , and x is a point outside of K. Define the distance from x to K by d(x,K) = inf {d(x,k) for k in K}. By considering the function f: K -> R given by f(k)= d(k,x), prove that the distance is positive and that there exists a point k in K such that d(x,k) = d(x,K).
5: Suppose K and H are two disjoint compact subsets of a metric space. Give a suitable definition of the distance between K and H, and prove that it is positive.
6: Show that the intersection of two compact subsets of a Hausdorff space is compact.
7: Show that the square with its sides identified in pairs (in the usual way, via translations) is homeomorphic to S1 x S1.
Homework 4, Feb 19; due Feb 27
1: Let X be a space. Define an equivalence relation on X by saying that x, y are equivalent if every continuous function f: X -> 2 satisfies f(x)=f(y). Show that the equivalence classes form a partition of X into open subsets (called "connected components" of X) and that X is connected if and only if there's just one class.
2: What are the connected components of the set of rational numbers Q (given the usual topology coming from the metric on R)?
3: Prove that the circle is not homeomorphic to the closed unit interval.
4: Prove that a connected subset of a space X must lie completely inside exactly one of the connected components of X.
5: Prove that if A, B are connected subsets of X which intersect non-trivially then the union A U B is connected.
6: Prove that any quotient of a connected space is connected, and likewise that the product of two connected spaces is connected.
7: Let X be the subspace of R2 consisting of all points (x, sin(1/x)) (where x is not equal to 0) as well as the single point (0,0). Show that X is connected but not path-connected. (Hint: for the second part, suppose there is a path from (-1, sin (-1)) to (1, sin(1)), consider what the projection of its image onto the x-axis must be, and derive a contradiction.)
Homework 5, Feb 27; due March 5
1: Show that the punctured plane (R2 minus the origin) is homotopy equivalent to the circle.
2: Show that the torus minus a point is homotopy equivalent to the figure-of-eight space (two circles joined at a point).
3: Classify the letters of the alphabet up to homotopy equivalence (and compare with the classification up to homeomorphism!)
4: Show that any map X -> S2 which is not surjective is homotopic to a constant map.
5: Prove the Lebesgue covering theorem: that if {Ui} is an open cover of a compact metric space X, then there exists a constant delta, such that any set whose diameter is less than delta lies in one of the Ui's. (The diameter of a set A is the supremum of the set of distances d(x,y) for x,y in A.)
6: Show how this theorem can be used to chop a path I -> S2 into pieces, each of which either does not hit the South pole, or does not hit the North pole. Use this to show that the fundamental group of S2 is trivial.
7: Prove that the fundamental group of X x Y, based at (x0, y0) is isomorphic to the product of fundamental groups of X and Y (with the obvoius basepoints).
8: Show that the fundamental group G of the Klein bottle is not abelian by finding a homomorphism from G onto the (non-abelian) dihedral group with 6 elements.