The Theory of Sets of Points

The Theory of Sets of Points, originally published in 1906, is now available as part of the Cambridge Library Collection of photographically-reproduced books. (For more on this series, see our review of a previous volume.) Written by William Henry Young and Grace Chisholm Young, this book represents an early attempt to organize a series of results related to sets of points. Today, we would describe it as a blend of general topology, set theory, and measure theory, all applied to subsets of \(\mathbb{R}\) and \(\mathbb{R}^2\). All of those subjects developed further after 1906, so no one would want to learn the subject from this book; instead, it is most valuable as a historical artifact.

For a modern reader, there are many places in the book where things seem “off.” On page 5, for example, the “number \(\infty\)” is introduced. What the authors in effect do is to compactify the line by adding a point at infinity. The result is that the distinction between closed sets and compact sets is completely erased, and one gets such theorems as “any nested sequence of closed sets has non-empty intersection.” Similarly, the Youngs say “content” where we would say “measure,” though they do define measurable sets in the modern sense.

The Youngs like closed sets, and many parts of the book build the theory first for closed sets and only afterwards for what they call “open sets,” i.e., sets that are not closed. This is probably the place where the modern reader is most likely to get in trouble: “open” does not mean open!

The most interesting part of the book for today’s reader is probably the discussion of subsets of the plane. I enjoyed the many interesting examples of space-filling and “crinkly” curves and the discussion of just how much can go wrong in a two-dimensional setting.

One gets a sense of what was considered difficult at the time from the chapter on “content.” The Youngs start by carefully developing a theory of content for intervals, then for disjoint unions of intervals, then for closed sets. Once past that boundary, they define outer and inner “content” for non-closed sets and use them to define measurable sets. Then we read

From the point of view of an exhaustive classification of open sets, it remains to be shewn whether sets other than measurable sets exist. This point is still open to question.

Indeed, the Youngs display great caution on any point where the Axiom of Choice might be needed. The axiom itself is only mentioned in footnotes and in an extended note in the appendix, presumably because it was still a matter of debate. But its consequences are almost always described as either unknown or still under debate.

The Youngs actually make an attempt to quantify how choice-dependent a proof actually is. The say that two sets are “simply equivalent” if there “a particular one-to-one correspondence connecting them.” (They mean, I think, that a single canonical function can be defined.) Then they say that two sets are \(\mu\)-fold equivalent if instead one can prove the existence of \(\mu\) one-to-one correspondences without singling out a canonical one. “Logically, multiple equivalence means less than simple equivalence,” they argue. But they do not push the idea much further than this, and only pick it up again in the appendix, in a note entitled “On simple and multiple equivalence and the mathematical law of arbitrary choice.”

In summary, this well-produced book gives us access to an interesting snapshot of a moment when point-set topology was a hot subject in full development.

Fernando Q. Gouvêa thinks old books are fascinating.