Mathematics 121 Final Terence Tao June 10, 1997 Problem 1. A set X ⊂ R^{n} is said to be star-shaped at the origin if for every x ∈ X, the line segment {tx : 0 ≤ t ≤ 1} is also contained in X.

Show that if X is star-shaped at the origin, then X is simply connected. (Hint: look at loops through the origin).

Problem 2. Let K, L be disjoint compact sets in a normal topological space X. Suppose that f : K → R and g : L → R are bounded and continuous functions on K, L respectively.

Show that there exists a bounded continuous function h : X → R such that h(x) = f(x) for all x ∈ K and h(x) = g(x) for all x ∈ L.

Problem 3. A topological space X is said to be locally separable if for every point x ∈ X there is a countable set E in X such that x is in the interior of \overline{E}.

Show that every compact, locally separable space is separable.

Problem 4.

(a) Show that the continuous image of any connected set is connected.

(b) Let {X_{α}}_{α∈A} be a collection of topological spaces. Suppose ∏_{α∈A} X_{α} is connected. Show that eaach X_{α} is connected.

Problem 5. Let X be a topological space such that, for every x ∈ X, one can find an open neighbourhood U of x which is path-connected.

(a) Show that the path-connected components of X are both open and closed.

(b) Show that every connected component of X is a path-connected component, and vice versa.

Problem 6.

(a) Let X be a locally compact Hausdorff space, and let X ∪ {∞} be the one-point compactification of X. Suppose that X is connected and non-compact. Show that X ∪ {∞} is connected.

(b) Let A, B be connected subsets of a topological space X such that A ∩ \overline{B} is non-empty. Show that A ∪ B is connected.

Problem 7. Let E be a covering space of X with covering map p : E → X, and let γ be a path in X which starts at a and ends at b.

For each point e in the fiber p^{−1}(a), let α_{e} be the lift of γ starting at e, and let f(e) be the final point of α_{e} (i.e. f(e) = α_{e}(1)).

(a) Show that f is a bijection from p^{−1}(a) to p^{−1}(b).

(b) Show that if X is path-connected, then all the fibers of E have the same cardinality.

Problem 8.

(a) Let X, Y be topological spaces, and define an equivalence relation ∼ on X ×Y by defining (x_{1}, y_{1}) ∼ (x_{2}, y_{2}) if and only if x_{1} = x_{2}.

Show that X × Y/ ∼ is homeomorphic to X.

(b) Define an equivalence relation ∼ on [0, 1] × [0, 1] by defining (x_{1}, y_{1}) ∼ (x_{2}, y_{2}) if and only if x_{1} = x_{2} and y_{1} − y_{2} is an integer. Show that [0, 1] × [0, 1]/ ∼ is homeomorphic to the cylinder

C = {(x, y, z) : 0 ≤ z ≤ 1, x^{2} + y^{2} = 1}.

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