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 $[0,1)$) and have lots of ‘holes’. It is useful for constructing topological counterexamples.

Definition

Definition Define the subset $K \subset \mathbb{R}$ as $K = \{1/n | n \ge 1 \}$. Generate a topology on $\mathbb{R}$ 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 $\mathbb{R}$ and denote the corresponding topological space as $\mathbb{R}_K$ (DR: this is not standard, but I need a symbol for it).