HFD and HFC type spaces, with applications
HFD's and HFC's arose early in the study of S and L spaces, gave rise to variations (e.g., HFD\(_{\text{w}}\)'s and HFC\(_{\text{w}}\)'s) and have been intensely studied by Juhász and his colleagues in Budapest. They are subspaces of Cantor cubes \(2^{\kappa}\) with strong combinatorial properties designed to give rise to interesting topological properties. This survey paper discusses their properties, relates them to other combinatorial principles, and studies the consequences of their properties for topology and vice versa (for example, the existence of 0-dimensional Hausdorff spaces in which every large subspace is separable but there is a small non-separable subspace; or spaces with properties similar to Ostaszewski spaces). To give the reader a taste of what this is about: a space \(X \subset 2^{\kappa 1}\) is HFD iff for every countable \(A \subset X\) there is a countable \(B \subset \kappa\) such that \(A|_{\kappa \setminus B}\) is dense in \(2^{\kappa \setminus B}\). Basic results are: HFD's are hereditarily separable, have no non-trivial convergent sequences, and are hereditarily collectionwise normal.
scientific article; zbMATH DE number 1891209
scientific article