nLab
K-topology

Idea

The K-topology is a topology on the reals $ℝ$ 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 $[0,1)$) and have lots of ‘holes’. It is useful for constructing topological counterexamples.

Definition

Definition Define the subset $K\subset ℝ$ as $K=\{1/n\mid n\ge 1\}$. Generate a topology on $ℝ$ by taking as basis all open intervals $(a,b)$ (these are the usual basic open sets) and all sets of the form $(a,b)-K$. Clearly only those open intervals that contain $(0,\frac{1}{2})$ need to be included in this latter collection. This topology is the K-topology on $ℝ$ and denote the corresponding topological space as ${ℝ}_K$ (DR: this is not standard, but I need a symbol for it).

Application

The set of rational numbers, with the subspace topology inherited from the inclusion $\mathbb{Q}\hookrightarrow {ℝ}_K$, 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.