The topological Hawaiian earring group does not embed in the inverse limit of free groups.
Self-homeomorphisms of the 2-sphere which fix pointwise a nonseparating continuum
We prove that the space of orientation preserving homeomorphisms of the 2-sphere which fix pointwise a nontrivial nonseparating continuum is a contractible absolute neighborhood retract homeomorphic to the separable Hilbert space .
Multiplication is Discontinuous in the Hawaiian Earring Group (with the Quotient Topology)
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.
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