Alexandroff line

If this isn’t one of Ben Elliott and Hunter Spink’s bizarre mathematical objects of the week, it certainly deserves to be. This is a strange object which locally behaves like the real line, but is much longer. Instead of considering the entire real line, we’ll focus on the positive half-line:

As shown above, it’s obtained by gluing together a countable set of intervals. In the diagram above, they are indexed by the ordinal ω. If we do the same thing for the ordinal ω², then we obtain the same ordinary real half-line:

The fusible numbers give a way of gluing together ε_0 intervals to obtain the real half-line. We can find similar constructions for all countable ordinals. Can we find an uncountable well-ordered subset X of the real line? The answer is no. For every x in X, there is a unique successor, S(x). Between x and S(x), there exists a rational number f(x). Clearly, the function f must be injective, so X is at most countably infinite.

So, what happens if we glue together ω_1 intervals, where ω_1 is the first uncountable ordinal? This must be longer than the real line, but for every point α, the interval [0,α) is homeomorphic to the real half-line [0,∞), which is in turn homeomorphic to the semi-open interval [0,1). Hence, we have something that locally resembles the real half-line, known as the Alexandroff ray. Gluing together two of these, back-to-back, gives the Alexandroff line, which locally resembles the real line despite being infinitely longer in both directions.

And guess what? In the Cartesian product (Alexandroff line)^2, we can pack uncountably many figure-eights. Can we pack as many figure-eights into the Alexandroff plane as we can circles? The answer, of course, is ‘if and only if the continuum hypothesis is true’…