nLab
K-topology

Context

Topology

topology (point-set topology)

see also algebraic topology, functional analysis and homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Basic homotopy theory

Contents

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)[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)(T_2) but not normal (T 4T_4).

Definition

Definition

Write

K{1/n|n 1} 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 βP()\beta \subset P(\mathbb{R}) on \mathbb{R} consisting of all the open intervals as well as the complements of KK inside them:

β{(a,b),|a<b}{(a,b)\K,|a<b}. \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 τ βP()\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

K(,τ β}. \mathbb{R}_K \;\coloneqq\; ( \mathbb{R}, \tau_{\beta}\} \,.

Properties

Proposition

The real numbers with their K-topology (def. 1) K\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 K\mathbb{R}_K. It remains to see that K\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

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

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

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

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

But every open neighbourhood of {0}\{0\} contains at least (ϵ,ϵ)\K(-\epsilon, \epsilon) \backslash K for some positive real number ϵ\epsilon. There exists then n 0n \in \mathbb{N}_{\geq 0} with 1/n<ϵ1/n \lt \epsilon and 1/nK1/n \in K. An open neighbourhood of KK needs to contain an open interval around 1/n1/n, and hence will have non-trivial intersection with (ϵ,ϵ)(-\epsilon, \epsilon). Therefore {0}\{0\} and KK may not be separated by disjoint open neighbourhoods, and so K\mathbb{R}_K is not normal.

Example

The set of rational numbers, with the subspace topology inherited from the inclusion K\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.

Revised on April 30, 2017 13:47:18 by Urs Schreiber (92.218.150.85)