Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)
Paul Fabel
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011), p. 77-83 / Harvested from The Polish Digital Mathematics Library
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The natural quotient map q from the space of based loops in the Hawaiian earring onto the fundamental group provides a naturally occuring example of a quotient map such that q × q fails to be a quotient map. With the quotient topology, this example shows π₁(X,p) can fail to be a topological group if X is locally path connected.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:281130 
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Paul Fabel. Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology). Bulletin of the Polish Academy of Sciences. Mathematics, Tome 59 (2011) pp. 77-83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba59-1-9/