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General Topology

Waclaw Sierpiński
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

This is a wonderful bit of mathematical history. Waclaw Sierpiński, known for his work in number theory, set theory, and topology, was one of the scholars of the Polish School that had its heyday in the early part of the twentieth century. Today even mathematical amateurs know his name in connection with fractal geometry: witness the Sierpiński triangle.

The book under review dates to 1948, its translation, reprinted by Dover, to 1956, and it is something of an extension of Sierpiński’s earlier (1934) Introduction to General Topology and contains rather more material that its predecessor. Specifically, Sierpiński spends a lot of time first on Fréchet spaces (each element of the space has an open neighborhood). Topological spaces, here, are characterized as Fréchet spaces satisfying additional axioms, and only then does Sierpiński go on to such mainstays as separable spaces (where we have countable dense subsets), Hausdorff spaces (with the first axiom of countability in place), normal spaces, and finally metric spaces and complete spaces.

Thus, the full sweep of what one needs to encounter in a serious first course in topology is present, but perhaps in a somewhat different form than one might encounter in more modern treatments. This is certainly of historical interest in that it is important to know the evolution of a mathematical subject in order to acquire a deeper and more holistic perspective on it; beyond this, however, it is useful to go at the indicated themes in a more or less “from the ground up” manner, getting one’s hands dirty building constructs of progressively greater relevance to the mainstream needs of mathematics. The sequence of topics mentioned above, covered by Sierpiński in the indicated order, reveal this trend: after all, the vast majority of us spend most of our mathematical lives in a Hausdorff and/or metric environment, for instance.

All this having been said, the coverage of the material in this old-fashioned book is also old-fashioned, which, to my mind is both refreshing and pedagogically sound (its countercultural quality notwithstanding). Sierpiński provides a great number of exercises (a.k.a. examples) throughout the text: a number of them are stated as problems or assertions but are immediately followed by their solutions (or proofs) — the responsible student obviously goes at them first before looking at what Sierpiński provides. Other exercises are void of such resolutions, although Sierpiński does provide the occasional hint. Thus, it’s not a pre-chewed and pre-digested business at all: the reader is asked to work his way through the text under Sierpiński’s tutelage and guidance, as it were. I guess R. L. Moore would approve.

Finally, Sierpiński is keen on stressing the axiomatic framework of his presentation, going so far as to say that in the orbit of his book’s chapters “new axioms are introduced about the space under consideration and theorems are derived from them. In general, the theorems of each of these chapters are not true in a space satisfying only the axioms of the preceding chapters.” This is clearly an extremely beneficial object lesson for anyone learning mathematics, especially topology, where things aren’t always what they seem: after all what can one say about a subject where such things as Milnor’s exotic spheres occur?

Thus, Sierpiński’s General Topology scores on a number of counts. It is of historical significance, it is a sound pedagogical work (you’ll learn topology very well, even if some of the language and notation is dated), and is an exemplar of high-level scholarship by none other than a premier member of the Polish School. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

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