nLab
line with two origins

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

What is called the line with two origins is the topological space which results by “gluing” a copy of the real line identically to itself, except at the origin. This is a basic example of a non-Hausdorff topological space (even a non-Hausdorff smooth manifold) and of the fact that quotient topological spaces of Hausdorff topological spaces need not be Hausdorff themselves.

Definition

Consider the disjoint union \mathbb{R} \sqcup \mathbb{R} of two copies of the real line \mathbb{R} regarded with its metric topology. Moreover, consider the equivalence relation on the underlying set which identifies every point x ix_i in the iith copy of \mathbb{R} (i{0,1}i \in \{0,1\}) with the corresponding point in the other, the (1i)(1-i)th copy, except when x=0x = 0:

(x iy j)((x=y)and((x0)or(i=j))). \left( x_i \sim y_j \right) \;\Leftrightarrow\; \left( \left( x = y \right) \,\text{and}\, \left( \left( x \neq 0 \right) \,\text{or}\, \left( i = j \right) \right) \right) \,.

The quotient topological space

()/ \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim

by this equivalence relation is called the line with two origins.

Properties

The line with two origins is a non-Hausdorff topological space:

Because by definition of the quotient space topology, the open neighbourhoods of 0 i()/0_i \in \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim are precisely those that contain subsets of the form

(ϵ,ϵ) i(ϵ,0){0 i}(0,ϵ). (-\epsilon, \epsilon)_i \;\coloneqq\; (-\epsilon,0) \cup \{0_i\} \cup (0,\epsilon) \,.

But this means that the “two origins” 0 00_0 and 0 10_1 may not be separated by neighbourhoods, since the intersection of (ϵ,ϵ) 0(-\epsilon, \epsilon)_0 with (ϵ,ϵ) 1(-\epsilon, \epsilon)_1 is always non-empty:

(ϵ,ϵ) 0(ϵ,ϵ) 1=(ϵ,0)(0,ϵ). (-\epsilon, \epsilon)_0 \cap (-\epsilon, \epsilon)_1 \;=\; (-\epsilon, 0) \cup (0, \epsilon) \,.

The Hausdorff reflection

H:TopTop Haus H \;\colon\; Top \longrightarrow Top_{Haus}

of the line with two origins is the real line itself:

H(()/) H\left( \left( \mathbb{R} \sqcup \mathbb{R} \right)/\sim \right) \;\simeq\; \mathbb{R}

However, it is T1 and sober.

It is also a topological manifold, even a smooth manifold, except that one often requires such to be Hausdorff to rule out examples such as this. In a similar vein, it is locally compact (and a compact version can be made by starting with [1,1][{-1,1}] instead of \mathbb{R}), paracompact, and so forth.

References

Revised on April 14, 2017 17:46:15 by Toby Bartels (108.167.41.14)