Question

New problems for 2020 1. A topological space is called a T3.space if it is a T, space and for every pair («,F), where € X and F(carefull), there is a continuous function 9 :X (0,1 such that f(x) 0 and f =1 on F. Prove that such a space has the Hausdorff Separation Property. (Hint: One point subsets are closed.] 2. Let X be topological space, and assume that both V and W are subbases for the topology. Show that the set of all intersections VanW8 -- where V EV and WA EW - also forms a subbase for the topology. If the two subbases are bases, show that the set of all intersections also gives a base for the topology. 3. Let (X, d) be a metric space, and define a new function d* on X X X such that d(u, v) = minimum{d(u,y), 1}. Verify that do defines a metric on X, and show that both metrics have the same open sets. (Hint: Given h > 0, show that the sets NeC) for € <h form a base for the topology.) 4. Let X be a topological space, and assume that for some pair of points p,9 € X there is a sequence of connected open subsets Vo, UL, U, such that pe Uo9 € U, and the consecutive intersections in Uit1 are all nonempty. Prove that there is a connected subset of X containing both p and q. (Hint: Show this is true for UnUin... Und 5. Recall that a space X is dense-in-itself if every point of X is a limit point. Prove that if A and B have this property then so does the product Ax B in the product topology. If X has this property and Y is a subspace of X, does Y also have this property? Either prove this or give a counterexample. 6. A topological space is said to be totally disconnected if there is a base U for the topology such that each V, in U is both open and closed. Prove that the set of rational numbers is totally disconnected. (Hint: Find a collection of closed intervals (ai, 6i] in R such that laj, 6] nQ = (a, b) n Q.)

Answer 1

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