Fixed Point Theorem and Closed Unit Ball in Euclidean Space
The Brouwer Fixed-Point Theorem Let denote the closed unit ball in Euclidean space : . Any continuous map from onto itself has at least one fixed point, i.e. a point such that . Proof Suppose has no fixed points, i.e. for . Define a map , , by letting be the point of intersection of and the ra ...continues
Covering Spaces : Compact Hausdorff Spaces and Homomorphisms
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.
(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.
Give the order and describe a generator of the group G(GF(729)/ GF(9)).
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 fr ...continues
sierpinski space is contractible
Let X be Sierpinski space: X={x,y} with topology {X,empty set, {x}} . prove that X is contractible.
Prove that "having the same homotopy type" is an equivalence relation on the set of topological spaces.
Partially-Ordered Sets ( Posets ) and Hom-sets
Show that a poset (partially-ordered set) is the same thing as a category in which all Hom-sets have at most one element.
Linearly Independent Subsets : Partially Ordered by Inclusion
Let X be any vector space over the field F, let L be a linearly independent subset of X, and A be the set of linearly independent subsets of X containing L. Then A is partially ordered by inclusion - why does it follow?
Determine the structure of the homology group H_n(X), n >= 0, if X is (a) the set of rational numbers with their usual topology; (b) a countable, discrete set.
