Coloring connected Hausdorff spaces

Motivation. I stumbled over the following hypergraph coloring concept when reading about an old (and open) problem by Erdos and Lovasz. Let H=(V,E) be a hypergraph such that for all e\in E we have |e| > 1, and let Z \neq \emptyset be a set. Then a map c:V\to Z is said to be a (hypergraph) coloring if for all e\in E the restriction c|_e is not constant (that is, the vertices pertaining to any edge are colored with more than 1 color). Trivially, for any nonempty hypergraph with no singleton edges, the identity map \text{id}:V\to V is a coloring. The chromatic number of a hypergraph with no singleton edges is the least cardinal \kappa such that there is a coloring c:V\to \kappa.

Equivalent topological formulation. If (X,\tau) is a connected Hausdorff space such that |X| > 1, we can use this to color the associated hypergraph (X,\tau\setminus\{\emptyset\}). (Indeed we can apply it to any topological space without isolated points.)

We can reformulate this in topological terms in the following way: Let (X,\tau) be a connected Hausdorff space. We define the nowhere dense covering number \nu(X) to be the minimum cardinality of a set {\cal N} of nowhere dense subsets of X such that \bigcup {\cal N} = X. It is not hard to see that \nu(X) equals the chromatic number of the hypergraph (X,\tau\setminus\{\emptyset\}).

For many standard connected Hausdorff spaces X we have \nu(X) = 2. For instance, if X = \mathbb{R} with the Euclidean topology, let {\cal N} = \{\mathbb{Q}, \mathbb{R}\setminus\mathbb{Q}\}. It took some effort to see that there are connected Hausdorff spaces X with \nu(X) > 2.

Question. For which cardinals \kappa > 2 is there a connected Hausdorff space (X,\tau) such that \nu(X) = \kappa?

Many more natural questions arise in this context, such as how does \nu(\cdot) behave with topological products, and so on. So far I haven’t found a reference studying this concept of “Hausdorff space coloring”.


About dominiczypen

I'm interested in general topology, order theory, and graph theory. This link takes you to my preprints on arXiv.
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s