# Precompact dots

Let $M, N$ be metric spaces such that $M$ is connected and $N$ is complete and locally compact. Let $\mathcal{F}$ be a uniformly equicontinuous family of functions from $M$ to $N$ such that there exists $x_0\in M$ with the property $(\star)$ that $\mathcal{F}(x_0)$ is precompact. Prove that all points $x\in M$ have property $(\star)$.

# Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group $D_{2n}$ with $2n$ elements over $\mathbb{C}$. Classify all irreducible representations for $D_{2n}$ over $\mathbb{Q}$.

# Fundamental theorem of algebra…?

Let $C_{n}$ be the cyclic group of order $n$ and consider $\mathbb{C}C_n$ the group algebra. Prove that the number of solutions to the equation $x^k = x$ for $x\in\mathbb{C}C_n$ is $k^n$ for $k>1$.

# Separating the étalé space

We briefly recall the construction of the étalé space of a sheaf. Given a sheaf $F$ over a space $X$, the associated étalé space is a topological space $E$ which, as a set, is the disjoint union of the stalks of $F$, with the topology induced by the basis $B(s, U) = \{s_x : x\in U\}$, where $U$ is an open set in $X$, $s\in F(U)$ and $s_x$ is the germ at $x$ of the element $s$.

Given a topological space $X$, consider the sheaf of real-valued continuous functions on $X$, which we denote $C$. It is more or less easy to see that the associated étalé space is not Hausdorff iff the space $X$ has the following property $(\star)$:

There exist continuous functions $f,g:U\to\mathbb{R}$, where $U$ is an open set, and a point $x\in U$ such that the germs of $f$ and $g$ at $x$ are different, but nevertheless for any open neighborhood $V\subseteq U$ of $x$ there exists an open set $W\subseteq V$ such that $f|_W = g|_W$.

For instance, $\mathbb{R}$ has property $(\star)$, and it’s easy to see that $(\star)$ is inherited by subspaces and products which contain any $(\star)$ factor. This, for instance, shows that subspaces of $\mathbb{R}^n$ are spaces satisfying this property.

Is there any characterization of all topological spaces for which the étalé space associated with the sheaf of real-valued continuous functions is not Hausdorff?

# Infinite dimensional Schur

Let $A$ be an algebra over $\mathbb{C}$ and let $V$ be an irreducible representation of $A$ with at most countable basis. Then any homomorphism of representations $\phi:V\to V$ is a scalar operator.

# Euler & Lie

Any compact connected Lie group has Euler characteristic zero. What are the possible values for the Euler characteristic of a connected, but not necessarily compact Lie group?

# Your degree is a chance for expansion

Let $X$ and $Y$ be compact oriented smooth manifolds of the same dimension, with $Y$ connected. Let $f : X \to Y$ be a smooth function. Take $y \in Y$ a regular value of $f$ and, for $x \in f^{-1}(y)$, define $o(x)$ as $+1$ if $f$ preserves orientation at $x$ and as $-1$ if it reverses the orientation. We define the degree of $f$ as the number $d(f) = \sum_{x\in f^{-1}(y)}o(x)$.

Prove the following extension theorem: Assume $Y=S^k$ is a $k$-dimensional sphere and $X=\partial W$ is the boundary of a compact oriented $(k+1)$-manifold $W$. There exists a smooth extension $F:W\to Y$ of $f$ if and only if $d(f)=0$.