Texas A&M University Department of Mathematics Volodymyr Nekrashevych Fall 2015 Math 636 — Problem Set 2 Issued: 09.11 Due: 09.18 2.1. Show that a subspace of a Hausdorff space is Hausdorff. 2.2. Let X be the plane R2 with the basis of topology given by the usual open sets together with the sets {(x, y) : x2 + y 2 < a, y 6= 0} ∪ {(0, 0)}. Show that this space is Hausdorff but not regular. 2.3. Suppose that A is a connected subset of a topological space, and A ⊂ B ⊂ A. Show that B is connected. 2.4. Suppose that a metric space (X, d) is connected. Show that for every x, y ∈ X and > 0 there exists a sequence x0 = x, x1 , x2 , . . . , xn = y such that d(xi , xi+1 ) < for all i = 0, 1, . . . , n − 1. Is the converse true? 2.5. Erdős’ example. P Let `2 be the set of all sequences (x1 , x2 , . . .) of real 2 numbers such that ∞ n=1 xn < ∞. It is known that d((x1 , x2 , . . .), (y1 , y2 , . . .)) = P∞ 1/2 ( n=1 (xi − yi )2 ) is a metric on `2 . We denote by kwk, for w ∈ `2 , the distance from w to the constant zero sequence. Let X be the subset of `2 consisting of sequences of rational numbers. Show that X is totally disconnected. Consider the set V = {w ∈ X : kwk < 1}, and let U ⊂ V be a neighborhood of (0, 0, . . .). Show that U \U is non-empty, and hence X is not zero-dimensional. (Hint: find a sequence wk = (a1 , a2 , . . . , ak , 0, 0, . . .) ∈ U such that distance from wk to X \ U is not more than 1/k.) 2.6. Local connectivity. A topological space X is called locally connected if for every point x ∈ X and every open neighborhood U of x there exists a connected neighborhood V of x such that V ⊂ U . Give an example of a connected but not locally connected subset of R2 .