Examples of topological groups that are not metrizable.

Question

I am looking for examples of topological groups but most are metrizable likes real numbers etc. Then there are Lie groups that I am not familiar with. Are there any simple enough to understand examples of non metric topological groups, or topological rings? How about ordered square or long line, can they be turned into a topological group?

Answer 1

$\mathbb{R}^I$ for any uncountable index set $I$ (pointwise addition), likewise $\{0,1\}^I$ (pointwise addition mod 2, compact as well), $C(X)$ (the continuous functions on an uncountable, completely regular topological space $X$, again with addition, with pointwise convergence topology), etc.

The ordered square and the long line are not homogeneous (the end points are distinct) so cannot be made into a topological group. Or if your versions of them are homogeneous, then they cannot be topological groups because thy're first countable and not metrisable (Birkhoff's theorem tells us that a first countable topological group is metrisable, which explains why my examples need to be "big").

Answer 2

If you really just want some non-metrizable topological group without other properties, you can create some examples as follows:

Take some group $G$ with order $\text{ord}(G) \geq 2$ and equip it with the trivial topology (only the empty set $\emptyset$ and $G$ being open). As a topological space, $G$ cannot be metrizable then:

Suppose it is and let $d$ be a metric on $G$. As $\text{ord}(G) \geq 2$, we can choose $a,b \in G$ with $a \neq b$. If $\epsilon = d(a,b)$, then we have $b \not\in B_{<\epsilon}(a),$ the open ball around $a$ with radius $\epsilon$, which is a contradiction to $G$ being equipped with the trivial topology.

As every map into a space with trivial topology is continuous, $G$ is a topological group, which means we have a non-metrizable topological group.