Hjalmar Ekdal topology

In mathematics, the Hjalmar Ekdal topology is a special example in the theory of topological spaces.^[1]

The Hjalmar Ekdal topology consists of N^* (the set of positive integers) together with the collection of all subsets of N^* in which every odd member is accompanied by its even successor. Examples: {2}, {6, 9, 10}

If all such subsets are declared "open", the "closed" subsets are consequently those in which every even member is accompanied by its odd predecessor.

It is not compact, but it is locally compact, paracompact and second countable.

Name

As it was the only original example (#55) in Steen and Seebach's Counterexamples in Topology, it was named by the undergraduates who worked on it. According to John Feroe, now at Vassar College:

Since this was a group project among three professors and five students, we played with the idea of choosing a pseudonym as the author of the book. So the question was, if we were going to be someone, who should we be? I had just taken a course on Henrik Ibsen (this was, after all, at St Olaf College, a Minnesota college founded by Norwegian-American Lutherans and very true to its heritage which was my heritage as well for that matter). I had been particularly taken by the play The Wild Duck, whose main character is a man named Hjalmar Ekdal. Hjalmar is a pathetic fellow who is unaware that almost everything he has was provided for him—house, business, wife, even his child. He is also unaware that he is quite incapable of succeeding on his own.

So we decided to call ourselves Hjalmar Ekdal since one way to look at what we were doing was collecting the work and examples provided by others cataloging rather than creating. We put up a big sign in the library alcove where we worked reading, "This space reserved for Hjalmar Ekdal," and posted quotations from Hjalmar Ekdal, such as "I haven¹t quite solved it yet, but I¹m working on it constantly."

And although the resulting book carries the names of the supervising faculty as the authors, Hjalmar does live on in that during that summer we had formulated a new example, and as its creators had the right to name it the Hjalmar Ekdal Topology ironically enough the only original example in the book.^[2]

References

1. Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 507446.
2. John Feroe (2002-04-21). "Hjalmar Ekdal Topology". Newsgroup: sci.math.