MAT 325: Topology Professor Zoltan Szabo Problem Set 1
Rik Sengupta rsengupt @ princeton.edu February 7, 2010
1. Munkres, p. 83, problem 7 Consider the following topologies on R:
T1 = the standard topology, T2 = the topology of RK T3 = the finite compleme
MAT 325: Topology Professor Zoltan Szabo Problem Set 9
Rik Sengupta [email protected] April 24, 2010
1. Munkres, p. 366, problem 2 For each of the following spaces, the fundamental group is either trivial, infinite cyclic, or isomorphic to the fundam
MAT 325: Topology Professor Zoltan Szabo Problem Set 2
Rik Sengupta [email protected] February 7, 2010
1. Munkres, p. 101, problem 13 Show that X is Hausdorff if and only if the diagonal = cfw_x x : x X is closed in X X. Solution. We will show both d
MAT 325: Topology Professor Zoltan Szabo Problem Set 3
Rik Sengupta [email protected] February 24, 2010
1. Munkres, p. 118, problem 6 Let x1 , x2 , . . . be a sequence of the points of the product space X . Show that this sequence converges to the po
MAT 325: Topology Professor Zoltan Szabo Problem Set 4
Rik Sengupta [email protected] February 28, 2010
1. Munkres, p. 152, problem 5 A space is totally disconnected if its only connected subspaces are one-point sets. Show that if X has the discrete
MAT 325: Topology Professor Zoltan Szabo Problem Set 5
Rik Sengupta [email protected] March 6, 2010
1. Munkres, p. 178, problem 4 Show that a connected metric space having more than one point is uncountable. Solution. Let X be a connected metric spac
MAT 325: Topology Professor Zoltan Szabo Problem Set 6
Rik Sengupta [email protected] March 30, 2010
1. Munkres, p. 199, problem 3 Show that every order topology is regular. Solution. We first prove that every order topology is Hausdorff. So let (X,
MAT 325: Topology Professor Zoltan Szabo Problem Set 7
Rik Sengupta [email protected] April 7, 2010
1. Munkres, p. 218, problem 1 Give an example showing that a Hausdorff space with a countable basis need not be metrizable. Solution. RK suffices as a
MAT 325: Topology Professor Zoltan Szabo Problem Set 8
Rik Sengupta [email protected] April 13, 2010
1. Munkres, p. 334, problem 1 A subset A of Rn is said to be star convex if for some point a0 of A, all the line segments joining a0 to other points