# 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 $[fX,gY] = fg[X,Y]+f(Xg)Y-g(Yf)X$.