nLab
Hawaiian earring space

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

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

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).