Hawaiian earring space

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.

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

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

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

Every neighborhood of $(0, 0)$ 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)