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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?

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    $\begingroup$ The short 'intuitive' answer is that any finite portion of the Hawaiian earring is isomorphic to a finite wedge sum of circles, but the difference lies in the limiting process and the 'shrinking' of the individual earrings - in particular, every neighborhood of the origin contains the entirety of infinitely many circles in H, but there are neighborhoods of the 'wedge point'of the countable wedge sum that don't contain the entirety of any of its circles. $\endgroup$
    – Steven Stadnicki
    Commented Oct 4, 2011 at 4:43

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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.

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You have two very nice answers discussing the difference between the topologies on these spaces. However, I thought I'd mention one slightly higher-level difference between them. The fundamental group of the wedge of infinitely many circles is the free group on countable many generators, one for each circle. This is a rather uncomplicated countable group. The fundamental group of the Hawaiian earring, however, is truly bizarre. In fact, it is uncountable and has many rather complicated relations in it.

When I first learned about this, I was shocked that closed subsets of the plane could have uncountable fundamental groups.

A nice paper that discusses this (and contains a good bibliography of earlier work) is "The combinatorial structure of the Hawaiian earring group" by Cannon and Conner, which appeared in Topology and its Applications, Volume 106, Issue 3, 6 October 2000, Pages 225-271.

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    $\begingroup$ I don't know that I'd call the free group of infinite rank "uncomplicated"... $\endgroup$
    – D Wiggles
    Commented Dec 19, 2014 at 16:33
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    $\begingroup$ @DWiggles it's easy to write down all elements of the free group. For the Hawaiian Earring group, even coming up with a system to describe all group elements is difficult, though Cannon and Connor do that. $\endgroup$
    – Cheerful Parsnip
    Commented Apr 18, 2023 at 15:51
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The point at which all the circles meet has different neighborhoods in each of the spaces you mention. In the case of a wedge of circles, the "wedge point" has contractible neighborhoods, while the corresponding point in the Hawaiian earring has no contractible neighborhoods.

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Since the the Hawaiian earring is a closed, bounded subset of $\mathbb{R}^2$ it's compact. But a countably infinite wedge sum (or countable bouquet) of circles is not, as one can readily produce an open cover that admits no finite subcover.

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  • $\begingroup$ It would be more useful to produce such an open cover than to say that one can easily produce it. $\endgroup$
    – Georges Elencwajg
    Commented Jun 24 at 11:03
  • $\begingroup$ It's useful to leave something for the reader, but to be explicit: Form the wedge sum $W$ of countably many copies of $S^1$ by identifying in $S^1 \times \Bbb Z$ all points $(1, n)$, $n \in \Bbb Z$, and take the open cover to be the images under the identification of the open sets $(S^1 \setminus \{-1\}) \times \Bbb Z$ and $(S^1 \setminus \{1\}) \times \{n\}$, $n \in \Bbb Z$. $\endgroup$
    – Travis Willse
    Commented Jun 24 at 19:15
  • $\begingroup$ Your wedge sum is not exactly the Hawaiian earring as defined in the question but it is trivial to adapt your idea to that earring: just replace $\mathbb Z$ by $\mathbb N$. $\endgroup$
    – Georges Elencwajg
    Commented Jun 25 at 14:36
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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 of a wedge sum of countably many circles is generated by countably many elements so countable.

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