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.
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