Defining Topological Groups and Algebras

We want to define topological groups as having a compatibility axiom that intuitively says that the “group operation is continuous.” But the group operation is a multivariable function, so how would we define this? And what about for general topological algebras? (Here, “algebra” is meant in the universal algebra sense, not in the “vector space with bilinear multiplication” sense.)

There are two possible approaches.

Definition 1. We use the box or product topology to define multivariable continuity. Since it’s a finite Cartesian product either is fine.

Definition 2. For every $a \in G$ , the functions $f_{a}(x) = ax$ and $g_{a}(x) = xa$ are continuous.

Are these in fact equivalent? Let’s say we want to prove that they are. Hopefully we can do the same for general topological algebras (really it’s just proving something about continuous functions and the topologies, nothing about the algebraic structure needs to be assumed.)

Let’s rephrase this purely topologically.

Goal. If $f:T^{2} \rightarrow T$ , then $f$ is continuous in the box or product topology iff for all $t \in T$ we have that $l(x)= f(x,t)$ and $r(x)= f(t,x)$ are continuous.

For a general algebra (let’s say finitary for now), we’d want this formulation:

General Goal. If $f:T^{n} \rightarrow T$ , then $f$ is continuous in the box or product topology iff for all $t \in T^{n}$ we have that $l_{i}(x) = f\left( t_{1},\ldots,t_{i - 1},x,t_{i + 1},\ldots,t_{n} \right)$ are continuous for all $i$ .

This basically reduces to: for arbitrary topological spaces, partial function continuity is equivalent to multivariable function continuity.

But I’m pretty sure this is false, even just for good old $\mathbb{R}$ . Multivariable continuity implies partial function continuity there. But what about the other way?

When we look this up, we find basic counterexamples in real analysis. So we have in general Def 2 does not imply Def 1, but we also generally have Def 1 implies Def 2, hence Def 1 is strictly stronger. Anyway, looking at Wikipedia, for topological algebras (including topological groups) it seems that the standard definitions all use Def 1.

First written some time ago (not sure when) while studying topological groups, topological rings, and other kinds of topological algebras.

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