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My dad passed away in February, and since then I've been spending time with his past, since I no longer have a 'present' with him.

In 1967 he did his master's thesis at the University of North Dakota on the equivalence of convergence in nets (he used the term Moore-Smith spaces) and filters.

I remember my dad saying that he and his thesis advisor found the result in his thesis published elsewhere after it was accepted, and that it took a bit of effort for them to find it. But at the same time, a paper from 1955 says that the equivalence of nets and filters was already well-known at that time.

I guess I'm looking for the context of the thesis, historical and otherwise. For example, does the thesis show equivalence by a different approach than what was typically used to show equivalence?

The thesis is online here: Convergence in Topological Spaces

Rich Jensen
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1 Answers1

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The mentioned article from 1955 by Bartle says, "It is well-known that these theories are equivalent in the sense that there are no proofs attainable by one that cannot be reached by the other." This does not quite say that any formal equivalences are known results. Indeed, Bartle proves some of these without giving another reference for them.

That a filter base is a net under reverse inclusion and that the filter base converges to a point in a topological space if and only if each net of "selectors" converges to the point is Theorem 8 in

Birkhoff, Garrett. "Moore-Smith convergence in general topology." Annals of Mathematics 38.1 (1937): 39-56.

Birkhoff didn't use the name "filter base," though; he introduced the idea himself independently in a presentation in 1935.

The result that the family of sets containing some remainder of a net are filters, and that every filter can be obtained this way is Satz 1.3 in

Sonner, Hans. "Die Polarität zwischen topologischen Räumen und Limesräumen." Archiv der Mathematik 4.5 (1953): 461-469.

This result corresponds to Theorem 4.1. in your father's thesis. Sonner writes that this was proven first by Jürgen Schmidt, who (in the next paper) denied this and claims this was probably already known.

A result that identifies filter with the intersection of eventuality filters obtained by selectors is given in

Bruns, Günter, and Jürgen Schmidt. "Zur Äquivalenz von Moore‐Smith‐Folgen und Filtern." Mathematische Nachrichten 13.3‐4 (1955): 169-186.

The statement is very different, but the idea parallels Lemma 4.1 and Theorem 4.2 in your father's thesis. However, your father constructed essentially a single "maximal" net of this form.

Relevant to the question, Bruns and Schmidt write that there was wide vague agreement that filters and nets are equivalent without precise results actually relating them.

The same construction of a filter (base) from a net is given as Proposition 2.1.(a)

Bartle, Robert G. "Nets and filters in topology." The American Mathematical Monthly 62.8 (1955): 551-557.

To obtain directed sets from filters, Bartle also uses reverse inclusion in Proposition 2.3(a). There is a footnote right before Proposition 2.3 that gives a different way to associate a directed set and a net with a filter. This is exactly the version that your father used in his thesis. Bartle credits a referee for suggesting it and mentions that the corresponding eventuality filter is then the original filter, which is Theorem 4.2 in your father's thesis.

Michael Greinecker
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