# Foiled by multiple roots

Let $\Sigma_3$ be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of $\mathbb{C}^3$ by looking at the non-leading coefficients. Prove that $\mathbb{C}^3\setminus \Sigma_3$ is homotopy equivalent to $S^3\setminus K$, where $K$ is the trefoil knot.

# Spooky categories

Suppose $C$ is an abelian category and $B\subseteq C$ is a subcategory which is also abelian. Is $B$ a sub-(abelian category) of $C$?

# Categorical noetherian-ness

A ring $A$ is noetherian iff the category $_A\mathrm{mod}$ of finitely generated $A$-modules is a full abelian subcategory of the category $_A\mathrm{Mod}$ of all $A$-modules.

# 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 $(\star)$.

# 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?

# 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?

# Small rings

Any field that is finitely generated as a ring is a finite field.