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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.

rips
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    The point at which the circles are wedged is different. In the countable wedge of circles, that point has a contractible neighborhood. In the Hawaiian earrings that point does not have a contractible neighborhood. – J126 Apr 15 '15 at 15:58
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    This question has answers explaining the difference between the two spaces. This answer explains why the Hawaiian earring is not a CW-complex. – Brian M. Scott Apr 15 '15 at 15:59

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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.

Michael Hardy
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  • Instead of viewing countable infinite circle as rose with infinite petal can I view it as sequence of circle of decreasing radius one inside another and then glue them at a point? – rips Apr 15 '15 at 16:08
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    . . . and if you view it in the latter way, it's a Hawaiian earring. ${}\qquad{}$ – Michael Hardy Apr 15 '15 at 16:12
  • sorry I am little confused here.According to my construction Is it wedge of circles?then hawaiian ring and wedge of circles are homeomorphic right?I am missing something...I'll be happy if you help. – rips Apr 15 '15 at 16:17
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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.

Cheerful Parsnip
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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)

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Anubhav Mukherjee
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