TITLE: Wedge sum of circles and the Hawaiian Earring
QUESTION [23 upvotes]: 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?

REPLY [2 votes]: From a algebraic topological view-point you can see a big difference between the fundamental group of infinite wedge sum of circles(denoted by $Y$) and the Hawaiian earring (denoted by $X$)...
let $C_n$ be the circle of radius $1/n$...
consider the retraction $r_n : X \rightarrow C_n$ collapsing all $C_i$'s expect $C_n$ to the origin...
each $r_n$ induces a surjective homomorphism $r_n^* :\pi_1(X) \rightarrow \pi_1(C_n)\approx \mathbb{Z}$...
the product of $r_n^*$ is a homomorphism $R: \pi_1(X) \rightarrow \prod_\infty\mathbb{Z}$ to the direct
product (not the direct sum) of infinitely many copies of $\mathbb{Z}$, and $R$ is surjective since for every sequence of integers $x_n$ we can construct a loop $f:I\rightarrow X$ that wraps $x_n$ times the loop $C_n$ in the time interval $[\frac{n-1}{n},\frac{n}{n+1}]$...
and it is continuous since every neighbourhood of the base point in $X$ contains all but finitely many circles.
So $\pi_1(X)$ is uncountable...
On the other hand the fundamental group od a wedge sum of countably many circles is generated by countably many elements so countable.