OPEN PROBLEMS IN TOPOLOGY

16 Steprans / Steprans’ Problems [ch. 21.3. Definition. AfilterF on ω is idempotent if F is isomorphic to F 2 andit is homogeneous if F is isomorphic to F|X for each X ∉F.By assuming that X is a counterexample to Question 1.1 and consideringonly the first two levels it can be shown that there is an idempotent homogeneousfilter on ω.? 20.Question 1.3. Is there an idempotent, homogeneous filter on ω?As in the case of Question 1.2, not even a consistent solution to Question 1.3is known. In fact only one example of an idempotent filter on ω is known andit is not known whether this is homogeneous. Finally it should be mentionedthat the questions concerning Toronto spaces of larger cardinalities and withstronger separation axioms also remain open.? 21.? 22.Question 1.4. Is there some non-discrete, Hausdorff, Toronto space?Question 1.5. Are all regular (or normal) Toronto spaces of size ℵ 1 discrete?2. Continuous colourings of closed graphsSome attention has recently been focused on the question of obtaining analogsof finite combinatorial results, such as Ramsey or van der Waerden theorems,in topology. The question of graph colouring can be considered in the samespirit. Recall that a (directed) graph G on a set X is simply a subset ofX 2 . If Y is a set then a Y -colouring of G is a function χ: X ↦→ Y such that(χ −1 (i) × χ −1 (i)) ∩ G = ∅ for each i ∈ Y .Byagraph on a topological spacewill be meant a closed subspace of the product space X 2 .IfY is a topologicalspace then a topological Y -colouring of a graph G on the topological spaceX is a continuous function χ: X ↦→ Y such that χ is a colouring of G whenconsidered as an ordinary graph.2.1. Definition. If X, Y and Z are topological spaces then define Y ≤ X Zif and only if for every graph G on X, ifG has a topological Y -colouring thenit has a topological Z-colouring.Even for very simple examples of Y and Z the relation Y ≤ X Z providesunsolved questions. Let D(k) bethek-point discrete space and I(k) thekpointindiscrete space. The relation I(k) ≤ X D(n)saysthateverygraphonXwhich can be coloured with k colours can be coloured with clopen sets and ncolours. It is shown in Krawczyk and Steprāns [19∞] thatifX is compactand 0-dimensional and I(2) ≤ X D(k) holdsforanyk ∈ ω then X must bescattered. Moreover, I(k) ≤ ω+1 D(k) istrueforeachk and I(2) ≤ X D(3) ifX is a compact scattered space whose third derived set is empty. This is thereason for the following question.