1. For each of the statements below determine whether it is true or false, providing reasons for your answer. Let (X, T ) be a topological space and A,B ⊆ X be two compact subsets.
(a) Is A ∪ B compact ?
(b) If (X, T ) is Hausdorff, is A ∩ B compact ?
2. Let f : [a, b] → R be continuous.
(a) Prove that the map g : [a, b] → R2, x 7→ (x, f(x)) is also continuous.
(b) Prove that the graph of f is a compact subset of R2.
3. Let (X, T ) and (Y, S) be connected topological spaces. Show that X × Y is connected in the product topology.
4. Consider the subsets
X = {(x, y) : 0 ≤ x ≤ 1 , y ∈ {0, 1}} ∪ {(x, y) : 0 ≤ y ≤ 1 , x ∈ {0, 1}} and Y = X ∪ {(x, x) : 0 ≤ x ≤ 1}
of R2 with the standard topology from R2. Show that X and Y are not homeomorphic.
5. Consider the subset
Z = {(0, 0), (0, 1)} ∪ ∞, n=1 Ln
of R2 (with the standard topology from R2), where Ln = {(1/n, y) : 0 ≤ y ≤ 1} for all n ≥ 1. Let U be a non-empty subset of Z which is both open and closed in Z. Show that if U contains one of the points (0, 0) and (0, 1), then it contains the other as well.
6. Let RP2 be the real projective space as defined in lectures, and let π : S2 −→ RP2 be the natural projection. Consider R3 and R4 with the standard topologies, and let the function f : R3 −→ R4 be defined by f(x, y, z) = (x2 − z2, xy, yz, xz).
(a) Show that there exists a function g : RP2 −→ R4 such that f = g ◦ π on S2.
(b) Show that g is continuous and one-to-one.
(c) Show that g is a homeomorphism between RP2 and the subset Y = g(RP2) of R4.
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