Efimov’s Problem

In their memoir [1, page 54] Alexandroff and Urysohn asked “existe-il un espace compact (bicompact) ne contenant aucun point (κ)?” and went on to remark “La resolution affirmative de ce problème nous donnerait une exemple des espaces compacts (bicompacts) d’une nature toute differente de celle des espaces connus jusqu’à présent”. The ‘compact’ of that memoir is our countably compact, ‘bicompact’ is ‘compact Hausdorff ’ and a κ-point is one that is the limit of a non-trivial convergent sequence. A look through the examples in [1] will reveal a few familiar classics: the ordinal space ω1 and the corresponding Long line, the double circumference, the Tychonoff plank (in disguise), the lexicographically ordered square, and the Double Arrow space. The geometric nature of the constructions made the introduction of non-trivial convergent sequences practically unavoidable and it turns out that the remark was quite correct as we will see below. The question was answered by Tychonoff [20] and Čech [4] using the very same space, though their presentations were quite different. Tychonoff took for every x ∈ (0, 1) its binary expansion 0.a1(x)a2(x) . . . an(x) . . . (favouring the one that ends in zeros), thus creating a countable set { an : n ∈ N } of points in the Tychonoff cube [0, 1], whose closure is the required space. Čech developed what we now call the Čech-Stone compactification, denoted βX, of completely regular spaces and showed that βN, where N is the discrete space of the natural numbers, has no convergent sequences. A natural question is whether one has to go to such great lengths to construct a compact Hausdorff space without convergent sequences. This then is Efimov’s problem, raised in [10].