nLab K-topology

Idea

The K-topology is a topology on the set of real numbers $\mathbb{R}$ which is finer than the Euclidean metric topology, but such that the new open sets are ‘piled up’ on the positive side of 0 (and actually are all contained in the half-open interval $[0,1)$) and have lots of ‘holes’. It is useful for constructing counterexamples in topology.

For instance the K-topology is an example of a topological space that is Hausdorff $(T_2)$ but not normal ($T_4$).

Definition

Definition

Write

$$K \coloneqq \{1/n \,\vert\, n \in \mathbb{N}_{\geq 1}\} \subset \mathbb{R}$$

for the subset of natural fractions inside the real numbers.

Define a topological basis $\beta \subset P(\mathbb{R})$ on $\mathbb{R}$ consisting of all the open intervals as well as the complements of $K$ inside them:

$$\beta \;\coloneqq\; \left\{ (a,b), \,\vert\, a\lt b \in \mathbb{R} \right\} \,\cup\, \left\{ (a,b) \backslash K, \,\vert\, a\lt b \in \mathbb{R} \right\} \,.$$

The topology $\tau_{\beta} \subset P(\mathbb{R})$ which is generated from this topological basis is called the K-topology.

We may denote the resulting topological space by

$$\mathbb{R}_K \;\coloneqq\; ( \mathbb{R}, \tau_{\beta}) \,.$$

Properties

Proposition

The real numbers with their K-topology (def. ) $\mathbb{R}_K$ are a Hausdorff topological space which is not a regular Hausdorff space (hence in particular not a normal Hausdorff space).

Proof

By construction the K-topology is finer than the usual euclidean metric topology. Since the latter is Hausdorff, so is $\mathbb{R}_K$. It remains to see that $\mathbb{R}_K$ contains a point and a disjoint closed subset such that do not have disjoint open neighbourhoods.

But this is the case essentially by construction: Observe that

$$\mathbb{R} \backslash K \;=\; (-\infty,-1/2) \cup \left( (-1,1) \backslash K \right) \cup (1/2, \infty)$$

is an open subset in $\mathbb{R}_K$, whence

$$K = \mathbb{R} \backslash ( \mathbb{R} \backslash K )$$

is a closed subset of $\mathbb{R}_K$.

But every open neighbourhood of $\{0\}$ contains at least $(-\epsilon, \epsilon) \backslash K$ for some positive real number $\epsilon$. There exists then $n \in \mathbb{N}_{\geq 0}$ with $1/n \lt \epsilon$ and $1/n \in K$. An open neighbourhood of $K$ needs to contain an open interval around $1/n$, and hence will have non-trivial intersection with $(-\epsilon, \epsilon)$. Therefore $\{0\}$ and $K$ may not be separated by disjoint open neighbourhoods, and so $\mathbb{R}_K$ is not normal.

Example

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

Related concepts

• Dowker space