nLab
Hawaiian earring space

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 Hawaiian earring space is a famous counterexample in algebraic topology which shows the need for care in hypotheses to develop a good theory of covering spaces.

It is an example of a space which is not semi-locally simply connected.

Definition

The Hawaiian earring space is the topological space defined to be the set

n{(x,y) 2:(x1/2 n) 2+y 2=1/2 2n}\bigcup_{n \in \mathbb{N}} \{(x, y) \in \mathbb{R}^2: (x - 1/2^n)^2 + y^2 = 1/2^{2n}\}

endowed with subspace topology inherited from 2\mathbb{R}^2.

More abstractly: the Hawaiian earring space can be described up to homeomorphism as the one-point compactification of a coproduct (in Top; a disjoint union space) of countably many open intervals. This shows that the specific radii converging to zero (which was taken to be 1/2 n1/2^n above) don’t actually matter.

Every neighborhood of (0,0)(0, 0) has non-contractible loops inside it (this would not be the case under the CW topology, i.e., the evident quotient space of countably many circles with the coproduct or disjoint sum topology, making the result a countable bouquet of circles).

Viewed in terms of general topology, it would be hard to sell the Hawaiian earring as a genuinely “pathological space”: it is compact, Hausdorff, connected and locally path-connected, etc. It’s really through the lens of algebraic topology and particularly the classical theory of covering spaces that it seems strange (it is not a CW complex, it is not semi-locally simply connected, its fundamental group is hard to compute).

Revised on May 16, 2017 14:01:01 by Urs Schreiber (131.220.184.222)