nLab
K-topology

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological 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)