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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?

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    @AsafKaragila's comment is spot-on. The one point compactification is perhaps the most general way of talking about adding a point at infinity.2012-09-10

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To expand on Asaf's comment, any ordinal number $\alpha$ can be made into a topological space by giving it with the topology generated by open intervals $(\beta,\gamma)$. This is called the order topology on $\alpha$. In this topology, the limit points are just the limit ordinals.

This space is compact if $\alpha$ is a successor ordinal ($\alpha = \beta+1$ for some $\beta$) but not if $\alpha$ is a limit ordinal (e.g., $\alpha = \omega$.) If $\alpha$ is a limit ordinal then the one-point compactification of the order topology on $\alpha$ is homeomorphic to the order topology on $\alpha+1$ via the map that fixes the elements of $\alpha$ and maps the point at infinity to the point $\alpha$ itself.