On regular but not completely regular spaces

Abstract

We present how to obtain non-comparable regular but not completely regular spaces. We analyze a generalization of Mysior's example, extracting its underlying purely set-theoretic framework. This enables us to build simple counterexamples, using the Niemytzki plane, the Songefrey plane or Lusin gaps.

The Niemytzki plane is kappa-metrizable

“…Let us add that there are many results about κ-normal spaces, for example compare [6]. Also, there exist many examples of a completely regular space which is not κ-normal, e.g., the ones which can be built using a technique called the Jones' machine, compare [7] or [1].…”

On RC-spaces

“…There exist examples of T 1 -spaces with bases consisting of closed-open sets, i.e., examples of completely regular spaces with regular normal bases, which are not RCspaces. These examples are completely regular spaces with a one-point extension to a regular space, which is not completely regular, for example, spaces considering in [7] or [5], also counterexamples constructed by the method initiated in [4].…”

On regular $κ$-bounded spaces admitting only constant continuous mappings into $T_1$ spaces of pseudo-character $\leq κ$

“…Regular spaces on which every continuous real-valued function (or, more generally, spaces on which every continuous function into a given space Y ) is constant are of particular interest in general topology. Such spaces were constructed and investigated in [1,3,4,5,7,8,9,10,11,14,15,16,17]. For instance, a well-known result of Herrlich [8] states the following:…”