nLab
Hawaiian earring space

Contents

Idea

The Hawaiian earring space is a famous counterexample in 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.

Every neighborhood of (0,0)(0, 0) has non-contractible loops inside it (this would not be the case under the CW topology?, where the result would be a countable bouquet of circles).

Revised on June 5, 2011 22:45:48 by Sridhar Ramesh (67.180.31.171)