The surreal line is topologically disconnected—or is it?

The surreal line is topologically disconnected according to a natural conception of connectedness. Nevertheless, on another conception—attending to set/class distinction—the surreal line is connected.

We all know that the familiar real line ℝ is topologically continuous and connected—there are no holes or gaps at all.

But what does it mean exactly to say that the real line is continuous? What does it mean for a topological space to be connected? And once we answer that question, I should like to know: Is the surreal line continuous in the same way? Is the surreal line No topologically connected?

Think about it...

The answer will be a pleasant philosophical surprise, for we shall see that the surreal line is disconnected with respect to one natural conception of connectedness; yet on another account—one giving great attention to the set/class distinction as it arises for the concept of open class—the surreal numbers are revealed as connected after all.

Welcome to this latest installment in my series of essays on the surreal numbers. Find the other essays in the surreal-numbers tag. Readers who wish to review basic notions should look at The Surreal Numbers, which offers a gentle introduction to this amazing and fantastical structure.