# Question concerning the term “connected.”

In the last lecture, Crista and I wondered about what the origin is of the term “connected” in Hopf algebras. Our guess was that it came from topology.

Here is why I thought that the term might have originated from topology: When one studies singular homology in algebraic topology, one of the first results one can prove about the singular homology group $H_0$ is that if $X$ is non-empty and path-connected, then $H_0(X) \cong \mathbb{Z}.$ Informally, our space comes in a single piece.

However, recall that we say that a Hopf algebra is connected if $H_0 \cong \mathbb{K}.$ Note that $\mathbb{K}$ is a field, and we know that there are fields that are uncountable.  Hence, this notion of connectedness in algebra need not agree with the notion of connectedness in topology. (For instance, consider a graded Hopf algebra that has, say, a $\mathbb{C}$-bialgebra structure.) Additionally, $\mathbb{Z}$ is not a field.

This being said, then, does anyone know or suspect what the origin of the term “connected” might be?

For those of you interested in reading a popular reference in algebraic topology, professor Allen Hatcher from Cornell University has made freely available his textbook in the subject: http://www.math.cornell.edu/~hatcher/AT/ATpage.html. In particular, the result I am referring to can be found on page 109 of this textbook.

2. Homology doesn’t always have coefficients in $\mathbb Z$. You can define homology and cohomology over every ring $\mathbb K$, and then connected spaces satisfy $H_0\left(X, \mathbb K\right) \cong K$.
Homology doesn’t always have coefficients in $\mathbb Z$. You can define homology and cohomology over every ring $\mathbb K$, and then connected spaces $X$ satisfy $H_0\left(X, \mathbb K\right) \cong \mathbb K$.