Let be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of by looking at the non-leading coefficients. Prove that is homotopy equivalent to , where is the trefoil knot.

# Author: fmartin

# Spooky categories

Suppose is an abelian category and is a subcategory which is also abelian. Is a sub-(abelian category) of ?

# Categorical noetherian-ness

A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.

# Precompact dots

Let be metric spaces such that is connected and is complete and locally compact. Let be a uniformly equicontinuous family of functions from to such that there exists with the property that is precompact. Prove that all points have property .

# Separating the étalé space

We briefly recall the construction of the étalé space of a sheaf. Given a sheaf over a space , the associated étalé space is a topological space which, as a set, is the disjoint union of the stalks of , with the topology induced by the basis , where is an open set in , and is the germ at of the element .

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

*There exist continuous functions , where is an open set, and a point such that the germs of and at are different, but nevertheless for any open neighborhood of there exists an open set such that .*

For instance, has property , and it’s easy to see that is inherited by subspaces and products which contain any factor. This, for instance, shows that subspaces of 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.