Fixed point theorem

See file - please answer the question in bold at the bottom

Covering Spaces: Algebraic Topology problem

Assume X and Y are arcwise connected and locally arcwise connected, X is compact Hausdorff, and Y is Hausdorff. Let f: X-->Y be a local homeomorphism. Prove that (X,f) is a covering space.

Homology group

(See attached file for full problem description) --- Determine the structure of the homology group Hn(X), n  0, if X is (a) the set of rational numbers with their usual topology; (b) a countable, discrete set.

generator of the group

Give the order and describe a generator of the group G(GF(729)/ GF(9)).

Unit square

Is it possible to partition a unit square [0, 1] X [0, 1] into two disjoint connected subsets A and B such that A and B contain opposing corners? I.e., such that A contains (0, 0) and (1, 1), and B contains (1, 0) and (0, 1)? *----0 | | | | 0----* Evidently, A and B couldn't be path-connected because a path running from (0, 0) to (1, 1) would intersect a path running from (1, 0) to (0, 1). So what about connected, but not path-connected, subsets? (The topology on the square is simply assumed to be the topology it inherits as a subspace of euclidean R^2.)