A topology τ on a fixed nonempty set X is said to satisfy the weakly continuous representation property if every weakly continuous not necessarily total preorder on the topological space (X, τ) admits a continuous order preserving function. Such a property generalizes the well known continuous representation property of a topology τ on a set X (according to which every continuous total preorder...

Let P1, . . . ,Pn be properties of graphs. A (P1, . . . ,Pn)-partition of a graph G is a partition of the vertex set V (G) into subsets V1, . . . , Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1, . . . , n. A graph G is said to be uniquely (P1, . . . ,Pn)-partitionable if G has exactly one (P1, . . . ,Pn)-partition. A property P is called hereditary if every subgraph of ev...

In this paper we classify noetherian hereditary abelian categories satisfying Serre duality in the sense of Bondal and Kapranov. As a consequence we obtain a classification of saturated noetherian hereditary categories. As a side result we show that when our hereditary categories have no nonzero projectives or injectives, then the Serre duality property is equivalent to the existence of almost ...

A hereditary property of combinatorial structures is a collection of structures (e.g. graphs, posets) which is closed under isomorphism, closed under taking induced substructures (e.g. induced subgraphs), and contains arbitrarily large structures. Given a property P , we write Pn for the collection of distinct (i.e., nonisomorphic) structures in a property P with n vertices, and call the functi...

The main result is the following. Let f:X→Y be a continuous mapping of completely Baire space X onto hereditary weakly Preiss-Simon regular Y such that image every open subset resolvable set in Y. Then Baire. classical Hurewicz theorem about closed embedding rational numbers into metrizable spaces generalized to spaces.

We show that for every hereditary permutation property P and every ε0 > 0, there exists an integer M such that if a permutation π is ε0far from P in the Kendall’s tau distance, then a random subpermutation of π of order M has the property P with probability at most ε0. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to th...

The edit distance between two graphs on the same labeled vertex set is the size of the symmetric difference of the edge sets. The distance between a graph, G, and a hereditary property, H, is the minimum of the distance between G and each G′ ∈ H with the same number of vertices. The edit distance function of H is a function of p ∈ [0, 1] and is the limit of the maximum normalized distance betwe...

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n 7→ |Pn|, where Pn denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then |Pn| = 2 2/2, where r = r(P) ∈ N is the so-called ‘colouring number’ of P . However, their result...