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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 new example of a quotient map such that q x q fails to be a quotient map. This also settles in the negative the question of whether the fundamental group (with the quotient topology) of a compact metric space is always a topological group with the standard operations.

Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)