We are thus led to a closer investigation of series with respect to limits. This investigation was made by Cantor and formed the basis of his definition of continuity, although, in its simplest form, this definition somewhat conceals the considerations which have given rise to it. We shall, therefore, first travel through some of Cantor's conceptions in this subject before giving his definition of continuity.
Cantor defines a series as "perfect" when all its points are limiting-points and all its limiting-points belong to it. But this definition does not express quite accurately what he means. There is no correction required so far as concerns the property that all its points are to be limiting-points; this is a property belonging to compact series, and to no others if all points are to be upper limiting- or all lower limiting-points. But if it is only assumed that they are limiting-points one way, without specifying which, there will be other series that will have the property in question—for example, the series of decimals in which a decimal ending in a recurring 9 is distinguished from the corresponding terminating decimal and placed immediately before it. Such a series is very nearly compact, but has exceptional terms which are consecutive, and of which the first has no immediate predecessor, while the second has no immediate successor. Apart from such series, the series in which every point is a limiting-point are compact series; and this holds without qualification if it is specified that every point is to be an upper limiting-point (or that every point is to be a lower limiting-point).
Although Cantor does not explicitly consider the matter, we must distinguish different kinds of limiting-points according to the nature of the smallest sub-series by which they can be defined. Cantor assumes that they are to be defined by progressions, or by regressions (which are the converses of progressions). When every member of our series is the limit of a progression or regression, Cantor calls our series "condensed in itself" (insichdicht).
We come now to the second property by which perfection was to be defined, namely, the property which Cantor calls that of being "closed" (abgeschlossen). This, as we saw, was first defined as consisting in the fact that all the limiting-points of a series belong to it. But this only has any effective significance if our series is given as contained in some other larger series (as is the case, e.g., with a selection of real numbers), and limiting-points are taken in relation to the larger series. Otherwise, if a series is considered simply on its own account, it cannot fail to contain its limiting-points. What Cantor means is not exactly what he says; indeed, on other occasions he says something rather different, which is what he means. What he really means is that every subordinate series which is of the sort that might be expected to have a limit does have a limit within the given series; i.e. every subordinate series which has no maximum has a limit, i.e. every subordinate series has a boundary. But Cantor does not state this for every subordinate series, but only for progressions and regressions. (It is not clear how far he recognises that this is a limitation.) Thus, finally, we find that the definition we want is the following:—
A series is said to be "closed" (abgeschlossen) when every progression or regression contained in the series has a limit in the series.
We then have the further definition:—
A series is "perfect" when it is condensed in itself and closed, i.e. when every term is the limit of a progression or regression, and every progression or regression contained in the series has a limit in the series.
In seeking a definition of continuity, what Cantor has in mind is the search for a definition which shall apply to the series of real numbers and to any series similar to that, but to no others. For this purpose we have to add a further property. Among the real numbers some are rational, some are irrational; although the number of irrationals is greater than the number of rationals, yet there are rationals between any two real numbers, however little the two may differ. The number of rationals, as we saw, is
. This gives a further property which suffices to characterise continuity completely, namely, the property of containing a class of