The (countably infinite) wedge sum of circles is the quotient of a disjoint countable union of circles $\coprod S_i$, with points $x_i\in S_i$ identified to a single point, while the Hawaiian earring/infinite earring H is the topological space defined by the union of circles in the Euclidean plane $\mathbb{R}^2$ with center $(1/n, 0)$ and radius $1/n$ for $n = 1, 2, 3, …$.
In the definition of the countably infinite wedge sum of circles, it is not specified the size of circles, the points which to be identified to a single point etc. So we can take disjoint union of circles of radius $1/n$ and identify a point from each to a common single point to get the Hawaiian earring.
I couldn't understand the difference between these two topological spaces. Can someone explain more precisely the difference between these two spaces?
Best Answer
The point is that topology the Hawaiian earring inherits from $\mathbb{R}^2$ is not the topology of the wedge sum of the circles which make it up. In particular, any open neighborhood of the origin in the Hawaiian earring completely contains all but finitely many of the circles, which is clearly not the case for an infinite bouquet of circles.