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


Definition Define the subset KK \subset \mathbb{R} as K={1/n|n1}K = \{1/n | n \ge 1 \}. Generate a topology on \mathbb{R} by taking as basis all open intervals (a,b)(a, b) (these are the usual basic open sets) and all sets of the form (a,b)K(a,b)-K. Clearly only those open intervals that contain (0,12)(0,\frac{1}{2}) need to be included in this latter collection. This topology is the K-topology on \mathbb{R} and denote the corresponding topological space as K\mathbb{R}_K (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 K\mathbb{Q} \hookrightarrow \mathbb{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.

Revised on May 5, 2010 08:07:56 by Urs Schreiber (