The K-topology is a topology on the reals which is finer than the usual topology, but such that the new open sets are ‘piled up’ on the positive side of 0 (and actually are all contained in ) and have lots of ‘holes’. It is useful for constructing topological counterexamples.
Definition Define the subset as . Generate a topology on by taking as basis all open intervals (these are the usual basic open sets) and all sets of the form . Clearly only those open intervals that contain need to be included in this latter collection. This topology is the K-topology on and denote the corresponding topological space as (DR: this is not standard, but I need a symbol for it).
The set of rational numbers, with the subspace topology inherited from the inclusion , is an example of a non-regular, totally path-disconnected Hausdorff space. This space can be used to construct a quasitopological groupoid which isn’t a topological groupoid.