Difference between Wedge of countable infinite circle and Hawaiian ear ring?

Question

Hawaiian ear ring is the union of countable circles at points (0,1/n) with radius 1/n.It seems to me that wedge sum of countable infinite circle is same as Hawaiian ring.But I found that this not true. I am thinking wedge of countable infinite circle as take a big circle and the put all the rest of the circle subsequently inside one another attaching to a common point,then diagram looks like Hawaiian ring. I can't figure out what is the difference between them.Can anyone help me in this direction. My second question is that why Hawaiian ring is not a CW complex but wedge of countable infinite circle is a CW complex? Thank you.

Answer 1

http://en.wikipedia.org/wiki/Hawaiian_earring

BEGIN QUOTE

The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but those two spaces are not homeomorphic. The difference between their topologies is seen in the fact that, in the Hawaiian earring, every open neighborhood of the point of intersection of the circles contains all but finitely many of the circles. It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.

END QUOTE

I wrote part of that section myself after this question came up in a comments section here on stackexchange a few years ago.

Answer 2

another difference you can see if you try to compute their Fundamental Groups... one side the fundamental group of infinite wedge of circles is free product of infinite copies of $\mathbb{Z}$ which is basically countable...on the other hand fundamental group of hawaiian earring is uncountable. (Wedge sum of circles and Hawaiian earring)

Answer 3

The Hawaiian Earring's topology is the subspace topology thinking of it as embedded in the plane. As such, any neighborhood of the point of intersection will contain infinitely many concentric circles. On the other hand, looking at the infinite wedge of circles, there are neighborhoods of the wedge point that do not contain any of the circles.