In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Hjalmar Ekdal topology is a special example in the theory of topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s.

The Hjalmar Ekdal topology consists of N^* (the set of positive integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s) together with the collection of all subsets of N^* in which every odd member is accompanied by its even successor. Examples: {2}, {6, 10}, {4, 7, 8, 101, 102, 448}

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

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

, 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

Henrik Ibsen

Henrik Ibsen was a major 19th-century Norwegian playwright, theatre director, and poet. He is often referred to as "the father of prose drama" and is one of the founders of Modernism in the theatre...

 (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

The Wild Duck

The Wild Duck is an 1884 play by the Norwegian playwright Henrik Ibsen.-Plot:The first act opens with a dinner party hosted by Håkon Werle, a wealthy merchant and industrialist. The gathering is attended by his son, Gregers Werle, who has just returned to his father's home following a self-imposed...

, 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.