Definition:Hjalmar Ekdal Topology

Definition

Let $\tau$ be the topology defined on the (strictly) positive integers $\Z_{>0}$ as follows:

A subset $H$ of $\Z_{>0}$ is in $\tau$ if and only if $H$ contains the (immediate) successor of every odd integer that is also in $H$:

$\tau := \set {H \subseteq \Z_{>0}: 2 n - 1 \in H \implies 2 n \in H}$

Then $\tau$ is referred to as the Hjalmar Ekdal topology.

The topological space $T = \struct {\Z_{>0}, \tau}$ is referred to as the Hjalmar Ekdal space.

Also see

• Hjalmar Ekdal Topology is Topology

• Results about the Hjalmar Ekdal topology can be found here.

Source of Name

This entry was named for Hjalmar Ekdal.

Historical Note

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 has been 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 haven't quite solved it yet, but I'm 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.

-- John Feroe

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (next): Part $\text {II}$: Counterexamples: $55$. Hjalmar Ekdal Topology