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Topology I Midterm, due October 22 Name: Instructions: This is a 48-point, 90-minute test. You may use your textbook, but no other aids are allowed. Unless otherwise specified, Rn is equipped with the standard topology and subsets of Rn are equipped with the subspace topology. Good luck! 1 /12 2 /12 3 /12 4 /12 Total /48 = /100% 1. Let `∞ = {(ai )∞ i=1 | ai ∈ R, maxi |ai | < ∞} be the set of bounded sequences and let d be the metric ∞ d(a, b) = maxi |ai − bi | where a = (ai )∞ i=1 and b = (bi )i=1 . Show that the unit ball ∞ B = {(ai )∞ i=1 | ai ∈ [−1, 1]} ⊂ ` is not compact, but that every sequence in 1 1 ∞ C = {(ai )∞ i=1 | ai ∈ [− , ]} ⊂ ` i i has a convergent subsequence. 2. Let R` be the set of real numbers equipped with the topology generated by the basis B = {[a, b) | a < b}. • Suppose that ai ∈ R` for all i. Give a necessary and sufficient condition for (ai )∞ i=1 to converge to L. • Show that R` is totally disconnected — that is, that every connected component consists of a single point. 3. Let L = {(x, 0) | 0 ≤ x ≤ 1}. The flea on the comb is the subspace of R2 consisting of the union 1 X =L∪ , y 0 ≤ y ≤ 1 and n = 1, 2, . . . ∪ {p}, n where p is the point (0, 1). Show that X is connected but not path-connected. 4. Suppose that X is a compact topological space, that Y is Hausdorff, and f : X → Y is a surjective continuous map. Show that there is an equivalence relation ∼ on X such that (X/∼) ∼ =Y. Blank page for scratch work