Volume 1 / Part 5 — *215. STRETCHES.
Reading edition
*215. STRETCHES.
Summary of *215.
A stretch of a series is any piece taken out of it, and not having any gaps; that is, it is a class contained in the series, and containing all terms which come between any two of its terms. Thus it is defined as
We denote the class of stretches by "," where "" stands for "stretch" or "Strecke." A stretch which has no predecessors is a section of ; one which has no successors is a section of . The properties of stretches are chiefly important in connection with compact series. In discrete series, stretches are the same as intervals.
If is transitive, stretches of are the products of sections of and sections of , i.e. of upper and lower sections of (*215·16). If is connected, and is a lower section, an upper section, then if the two have a stretch in common, we have A slightly more general form of this proposition is
*215·165.
A specially important case is when and have just one term in common. In this case we have
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When has more than one term, if the upper limit or maximum of and the lower limit or minimum of both exist, the latter precedes the former (*215·52); if and have no common part, but together exhaust the field of , we have either or , assuming (*215·54). Hence if has no immediate successor, it must be identical with . Thus we have
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The above propositions will be useful in Section C (*231 and *233).
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