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

### Volume 59 / 2011

Bulletin Polish Acad. Sci. Math. 59 (2011), 77-83
MSC: Primary 54G20; Secondary 54B15.
DOI: 10.4064/ba59-1-9

#### Abstract

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 $q\times q$ fails to be a quotient map. With the quotient topology, this example shows $\pi _{1}(X,p)$ can fail to be a topological group if $X$ is locally path connected.