Undergrad Topology - 400 Level

1. Prove that v(Г) - e(Г) = 1 for any tree T. (v :vertices and e : edges) 2. Even better, show that v(Г) - e(Г) ≤ 1 for any graph Г, with equality precisely when Г is a tree.

Undergrad 400 level Topology.

Find a tree in the polyhedron of figure 1.3 which contains all the vertices. Construct the dual graph Г and show that Г contains loops. (You don't have to construct the graph, but please describe it to me how it looks like.) (SEE ATTACHMENT)

Topology

14. Make a Mà¶bius strip out of a rectangle of paper and cut it along its central circle. What is the result? 15. Cut a Mà¶bius strip along the circle which lies halfway between the boundary of the strip and the central circle. Do the same for the circle which lies one-third of the way in from the boundary. What are the resulting spaces? (Questions are also included in attachment)

Topology - Undergrad 400 Level

17. Define f: [0, 1) → C...Deduce that f is not a homeomorphism. (See attachment for full question)

Topological space

19. Let X be a topological space and let Y be a subset of X. Check that the so-called subspace topology is indeed a topology of Y. (question is also included in attachment)