© 1996, Topology Atlas

Ever since the Cech-Stone compactification betaX was discovered, topologists have been trying to `see' what it looks like, especially for fundamental spaces like N (the discrete space of natural numbers) and R (the real line). Of particular interest is of course the set of new points, that is the remainder X*=betaX minus X.

The remainder of N is now fairly well understood --- although several natural questions remain open, see [5]. Indeed, the known results on betaN and its remainder could fill several volumes. One of the reasons for this success is that the space N has no real structure so that in translating statements from the remainder to N one ends up with statements that are simply about sets and do not refer to any topological property. Therefore all the machinery of set theory can then be brought to bear on the problem at hand.

The remainder of R is the topological sum of the remainders of H=[0,infinity) and (-infinity,0] respectively. These remainders are homeomorphic, so we usually only deal with H and its remainder. The `problem' with H and its remainder is that both spaces are continua; this gives us much less room to maneuver. This does not mean that it is impossible to say something about our spaces. It has been established that the remainder of H is an indecomposable and hereditarily unicoherent continuum, but not hereditarily indecomposable. The basic structure of this remainder has also been determined. A reasonably up-to-date survey is [4]; some more recent results are in [1], [2] and [3].

To answer the question in the title: the real line is, as is the set of natural numbers, a fundamental object in mathematics. It is therefore of utmost importance to learn as much as we can about objects that are naturally associated with it.

We conclude with two (very) open problems about the remainder of H:

1. Give a topological characterization of the remainder of H --- in the spirit of Parovicenko's characterization of the remainder of N.
2. Determine the number (up to homeomorphism) of subcontinua of the remainder of H --- at present we know fourteen, in ZFC.

1. Alan Dow and Klaas Pieter Hart, Cech-Stone remainders of spaces that look like [0,infinity), Acta Universitatis Carolinae---Mathematica et Physica 34 (1993), no. 2, 31--39.
2. Alan Dow and Klaas Pieter Hart, Cut points in Cech-Stone remainders, Proceedings of the American Mathematical Society 123 (1995), 909--917.
3. Alan Dow and Klaas Pieter Hart, A new subcontinuum of betaR minus R, Technical Report 95-114, Faculty TWI, TU Delft, December 1995, Gzip'ed PostScript file available by anonymous ftp.
4. Klaas Pieter Hart, The Cech-Stone compactification of the Real Line, In: Recent Progress in General Topology (Miroslav Husek and Jan van Mill, eds.), North-Holland, Amsterdam, 1992, pp. 317--352.
5. Klaas Pieter Hart and Jan van Mill, Open problems on $\beta\omega$, In: Open Problems in Topology (Jan van Mill and George M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 97--125.