Topology Exercises

Point Set Topology

  • Show that the rationals \mathbb{Q} are not locally compact.
  • Show that a connected metric space having more than one point is uncountable.
  • Let f:S^{1} \to \mathbb{R} be a continuous map. Show there exists a point x of S^{1} such that f(x) = f(-x).
  • Give an example of a subset of \mathbb{R} having uncountable many connected components. Can such a subset be open? Closed?
  • A standard theorem states that a continuous real valued function on a compact set is bounded. Prove the converse: If K is a subset of \mathbb{R}^{n} and if every continuous real valued function on K is bounded, then K is compact.
  • An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.
  • Let S be the Sierpinski space (that is the space S = {0,1} with the topology determined by letting \varnothing, 1, and S be the open sets), and prove that S is the only space (up to homeomorphism) such that if X is any other topological space then continuous functions from X to S are in bijection with open sets of X.

Manifold Theory

  • Let \mathbb{R} be the real line with differentiable structure given by the maximal atlas of the chart (\mathbb{R},\phi(x) = x). Let \mathbb{R}' be the real line with differentiable structure given by the maximal atlas of the chart and (\mathbb{R},\psi(x) = x^{1/3}). Show that these two differentiable structures are distinct, and that there is diffeomorphism between \mathbb{R} and \mathbb{R}'.
  • Let the additive group 2\pi\mathbb{Z} act on \mathbb{R} on the right by x \cdot 2\pi n = x+2\pi n where n is an integer. Show that the orbit space \mathbb{R}/2\pi\mathbb{Z} is a smooth manifold.
  • Let F:\mathbb{R}^{2} \to \mathbb{R}^{3} be the map F(x,y) = (x,y,xy). Let p = (x,y) \in \mathbb{R}^{2}. Compute F_{\ast}(\partial/\partial x\mid_{p}).
  • Let G be a Lie group with multiplication map \mu:G \otimes G \to G and inverse map \iota:G \to G. Prove that the differential of \mu at the identity is addition of tangent vectors. Prove that the differential of \iota at the identity is negation of tangent vectors.
  • Let x, y, z, and w be the standard coordinates on \mathbb{R}^{4}. Is the solution set of x^{5}+y^{5}+z^{5}+w^{5} = 1 in \mathbb{R}^{4} a smooth manifold assuming the subset is given the subspace topology?
  • Let the unit sphere S^{n} in \mathbb{R}^{n+1} be defined by the equation \sum_{i = 1}^{n+1}(x^{i})^{2} = 1. If p = (p^{1},\ldots,p^{n+1}) \in S^{n}, prove that X_{p} = \sum a^{i}\,\partial/\partial x^{i}\mid_{p} is tangent to S^{n} if and only if \sum_{i = 1}^{n+1}a^{i}p^{i} = 0.
  • Show that a smooth map f from a compact manifold N to \mathbb{R}^{m} has a critical point.
  • Prove that if N is a compact manifold, then an injective immersion f:N \to M is an embedding.
  • Prove two C^{\infty} vector fields X and Y on a manifold M are equal if and only if for every C^{\infty} function f on M, we have Xf = Yf.
  • Let X be the vector field x\,d/dx on \mathbb{R}. For each point p \in \mathbb{R}, find the maximal integral curve c(t) of X starting at p.
  • If f and g are C^{\infty} fuctions and X and Y are C^{\infty} vector fields on a manifold M, show that [fX,gY] = fg[X,Y]+f(Xg)Y-g(Yf)X.