The point at infinity $\infty$ makes $\mathbb{R}$ into a topological circle, i.e. into the real projective space $\mathbb{R}P$: $\infty$ enhances the original structure ($\mathbb{R}$) and makes it into another structure (not ordered anymore).
In a similar setting the first limit ordinal $\omega$ enhances the original structure ($\mathbb{N}$) and makes it into another structure (still ordered).
Projective spaces are at the heart of algebraic geometry while limit ordinals are at the heart of set theory.
Both concepts - the point at infinity $\infty\ $ and the first limit ordinal $\omega$ - and their respective roles (played in their resp. theories) are not too hard to grasp, if taken each of its own.
For the outsider/novice they "feel" strongly related: "(the) something at infinity".
(For the visually inclined there is a strong and striking resemblance between the symbols $\omega$ and $\infty$ [= a two-sided $\omega$].)
Especially for the outsider/novice it may be difficult to comprehend the "deep" connection between these two concepts and their respective roles - if there is any.
If there is such a "deep" connection: How can it be comprehended in the most "easiest" way?
If there is no such "deep" connection: How can it be comprehended why this was a (conceptual) illusion?