By Dominic van der Zypen
The motivation for this post is that these days I am goofing around with lattices arising from subspaces of hypergraphs, which are complete and atomistic. Let be a lattice with a bottom element . We say that is an atom Continue reading
Motivation. We show that the space satisfies the selection principle , but not . This gives a negative answer to the question in the general setting. Below is a self-contained treatment of the matter. Let be a topological space. Continue reading
Given a set and , we let and . If is a topological space, the Vietoris topology on the ground set is the topology generated by the subbasis is and denoted by . Let Continue reading
Motivation. The three-dimensional cube can be formalized by where vertices are connected by an edge if and only if their symmetric difference contains exactly element. If we want to assign to each vertex one of the colors black and white Continue reading
Let denote the collection of infinite subsets of (the first infinite ordinal) and let be any map. We say that is monochromatic with respect to if restricted to is constant (or, equivalently, if for all with ). We say is Continue reading
Let denote the first infinite cardinal that is, the set of non-negative integers. Let be the smallest prime number, and let enumerate all prime numbers in ascending order. Let be a free ultrafilter on . We consider the field Continue reading
Suppose you want to have a graph with chromatic number equaling some value , such that is minimal with this property. So you end up with a -(vertex-)critical graph. It is easy to construct critical graphs by starting with some Continue reading
The starting point of this blog post is a slight reformulation of the separation axiom: A topological space is if for all there is a set such that Given a cardinal , we say that a space is if for Continue reading
For we write if is finite, and we write if and . A tower is a collection of co-infinite subsets of such that for all we have and either or . ( is co-infinite if is infinite.) If are towers, Continue reading
Many people I talk to about graph theory feel some uneasiness when it comes to the notion of minor . I want to try to alleviate this feeling by providing the definiton of minor that I work with. First an easy Continue reading