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Math 434/542 Fall 2015 Test 1, 9-28-15 Carter 1. Let X denote a set. Let P(X) = {A ⊂ X}. Suppose that T ⊂ P(X). Define the term T is a topology on X. 2. In case X is finite, say X has n elements. Indicate why P(X) has 2n elements. 3. Suppose that X is infinite. Let T = {A ⊂ X| X \ A is finite }. Here X \ A = {x ∈ X| x ∈ / A}. Show that T is a topology on X. (If you have trouble with this because you could not solve #1, please ask for the definition.) f 4. Suppose (X, T ) and (Y, V) are topological spaces and Y ← X denotes a function. Define f is continuous on X. 5. Describe the open sets in the usual topology on the real line R. This is the topology induced by the metric d(x, y) = |y − x| on R. 6. Consider the identity function on R. Also consider the usual and finite complement topology (problem #3). The identity function is continuous in one direction. Which one? 7. Let X denote a set; and A 7→ A a function from P(X) to P(X) such that (a) ∅ = ∅; (b) A ⊂ A; (c) A = A; (d) A ∪ B = A ∪ B. Show that T = {X \ A| A ∈ P(X)} is a topology on X. 1