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

摘要

Summary. 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 π_1(X,p) can fail to be a topological group if X is locally path connected.