Consider the smallest infinite ordinal ω. Notice that it's typeset upright by default to denote that the meaning of the symbol is fixed.
In normal text this causes no problems:
Now we present the specification of traffic lights as a machine with labeled transitions via regular and ω-regular expressions.
However, text (e.g., in titles) is often emphasized by making it italic. Therein, you can typeset italics ω:
Example 1.2.3 (Specification of traffic lights as a machine with labeled transitions via regular and ω-regular expressions) In the context of Example 1.2.1 consider […]
Or you can typeset upright ω:
Example 1.2.3 (Specification of traffic lights as a machine with labeled transitions via regular and ω-regular expressions) In the context of Example 1.2.1 consider […]
Independent of how folks do it when they are out of time and what various typesetting programs produce on various input, how should it be done correctly™ or properly™ and why? (The publisher's guidelines don't say anything on the matter.)
All languages, including technical language for mathematics, are determined by usage, and not by what any authority says. See the classic Descriptive vs Prescriptive approaches in linguistics. As such, there isn't a ''correct" answer, rather there are norms that people tend to follow, and these may change over time.
From what I've seen, it is customary to emphasize in upright text by italicizing, and to emphasize in italics text by making it upright. Also, I've noticed that numbers, functions, operators, and mathematical symbols in general are written in displaystyle, even when in they are in the middle of the text, which makes them upright.
So, for example, in your question I would typeset the $\omega$ upright in the text. But there isn't a ''proper" way to do it, just norms and personal preference.