I would like to verify that H(adsbygoogle = window.adsbygoogle || []).push({}); _{1}(Hawaiian Earrings) is uncountable. Let's call the Hawaiian Earrings X and build it as a quotient space of [0,1] by the set {0,1, 1/2, 1/3, 1/4, 1/5, ...}. The map [tex]f:[0,1] \rightarrow X [/tex] with f(x)=[1-x] is a continuous map from [tex]\Delta^1[/tex] into X. Explicitly, it hits every ring exactly once in the order of the radii of the rings. Similarly, every ordering [tex](n_i)[/tex] of the positive integers corresponds to a continuous map if the ith ring that the map traverses is the one of length [tex]1/n_i - 1/(n_1+1)[/tex], with 1 being mapped to 0.

It is easy to verify that this set of maps is uncountable. Now of course I have to verify that (ideally) none of the maps are homologous, and this is where I am stuck. It is straight forward to show that these maps are not pairwise homotopic, but I have no idea how to show that they are in different homology classes. Any hints or ideas?

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# Homology of Hawaiian Earrings

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