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Multiplication is discontinuous in the Hawaiian earring group (with the quotient topology)

Fabel P.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 1

77-83

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 x 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.