Talk:Hawaiian earring

From Wikipedia, the free encyclopedia

The article states: "It has been shown that $G$ embeds into the inverse limit of the free groups with $n$ generators, $F_n$, where the map from $F_n$ to $F_{n+1}$ is just the one sending the generators of $F_n$ to the first n generators of $F_{n+1}$."

This cannot be correct, as in the inverse limit, there is not a map from $F_n$ to $F_{n+1}$ but rather from $F_{n+1}$ to $F_n$. Indeed, the direct limit given by the natural maps from $F_n$ to $F_{n+1}$ is merely the free group on countably infinitely many generators.

When comparing the Hawaiian earring to the infinite wedge of circles $H'$, the article claims that $H'$ " is the one-point compactification of a countable disjoint union of open intervals." This can't be true, since the infinite wedge of circles is not compact (taking the interior of each 1-cell and a small neighborhood of the point at which they meet gives an infinite open cover with no proper subcover). It seems that this has also caused confusion in the article on Quotient Spaces, which refers to R/Z as the Hawaiian earring. BarrySimington 16:02, 8 March 2007 (UTC)