Problems:
Al. (a) State the Axiom of Choice.
(b)Define a well ordering. Can every set be well ordered?
(c)Give a well ordering for the rationals. Is the Axiom of Choice needed for this?
A2. (a) Let X be a nonempty set. Define what is meant by a topology T on X .
(b) Define a neighbourhood of a point x in (X, T).
(c) State what it means for a function f : X→Y, where X and Y are
topological spaces, to be continuous.
(d) Prove that f : X→Y is continuous if and only if for each open set H in Y,f-1(H) is open in X.
A3. (a) Define what it means for a topological space X to be a Hausdorff space (ie, a T2 space).
(b) Define the product topology on a product of topological spaces X = ∏αXα. Prove that if Xα is nonempty and each X, is Hausdorff, then X is Hausdorff.
(c) Define what it means for a topological space to be compact. Prove that every closed subset of a compact space is compact.
SECTION B
B4. (a) Define a metric space. What does it mean for two metrics to be equivalent? What does it mean for a topological space to be metrizable?
(b) Suppose (X, p) is a metric space. Define ρ˜ :X x X →R by ρ˜(x,y)= min{ρ(x,y),1} . Show that ρ˜ is a metric for X .
(c) For the metric space (X, p), let U p(x,r) be the open unit disk about the point x of radius r. Prove the following statement: Two metrics p and µ, for a set X are equivalent if and only if for each x ∈ X and each r > 0, there exist s, t > 0 such that Up(s, s) C Um(x, r) and Up(x, t) C Up(x, r).
(d) Use (c) to show that the two metrics in (b) are equivalent.
B5. (a) Define a homeomorphism between topological spaces X and Y. Define what is meant by a topological invariant.
(b)State what it means for a map f : X → Y to be open. Show that a continuous open bijection is a homeomorphism.
(c)(i) Recall that Fr E, the frontier of a subset E of a topological space X is defined as E‾ n (X-E)‾. Prove that E‾= E U Fr E.
(ii)A topological space is 0-dimensional if and only if whenever x E V and V is open, there is an open set U with empty frontier such that x∈U⊂V. Show that the rationals in the relative topology as a subset of HI with the usual topology is a 0-dimensional set.
(iii)Show that being 0-dimensional is a topological invariant.
B6. (a) Define what it means for a topological space to be connected.
(b)Suppose that A and B are subspaces of a topological space X, and that U⊂A∩B is open in both A and B in the relative topologies. Show that U is open in A U B in the relative topology.
(c)Suppose that X is connected and that A is a connected subset of X . Suppose further that X - A = U U V, where U and V are nonempty disjoint open subsets of X - A in the relative topology for X - A. Show that A U U is connected.
(d)(i) Define what is meant by a component of a topological space.
(ii)A space X is totally disconnected if and only if each of its components consists of a point. Show that Q in the relative topology as a subset of R with the usual topology is totally disconnected.
(iii)A connected space X is said to have an explosion point p if and only if X - {p} is totally disconnected. Find an example of such a space.