Differently named concepts that turn out to be the same

Question

(Apologies if this is a duplicate. Wasn't able to find a question with similar keywords)

Motivation: Recently started reading Willard, General Topology. In one of the first exercises, he defines a radial (sub)set (of $\mathbb{R}^2$) to be one such that every point of it has a line segment that contains it and is contained in that set for every possible direction. He then asks to prove that these sets form a topology and to compare it with the euclidean one. He then remarks that the plane with this topology is called the radial plane.

That remark seems to suggest that those two topologies aren't the same. And indeed they aren't. But if i was making a book i wouldn't want to put such information in exercises. So game theory tells me i'd have to balance my strategy by making some remarks like that in which different names are given to things that end up being the same.

Question: What are some examples of concepts in math that are defined differently$^{(*)}$, and whose definitions are given different names but end up being the same? Here's one to start:

• In ZF with (countable) choice, infinite sets and dedekind-infinite sets are the same.

(*) (i'm not sure how to define what defined differently should mean. We could easily construct trivial examples of definitions which are slightly different and end defining same concepts. And it also depends on context sometimes, ie with or without axiom of choice. If anyone has a good definition for this please tell, otherwise just rely on intuition to try avoid those trivial examples i guess.)

Answer 1

I feel like the Euclidean plane is something that has been defined in nontrivially different ways. I do not know if it suits you since I don't think it ever really had two names, but it did have more than one foundation.

One way is synthetically: you can use something like Hilbert's Axioms for ordered geometry to get something called the Euclidean plane.

Or you can do it analytically, first establishing axioms for the real numbers $\mathbb R$, and then defining the Eulcidean plane to be $\mathbb R\times \mathbb R$ with measures of length and angle.

The two starting points are (IMO) quite different from each other, but they arrive at the same structure.

Another thing that brings to mind are the many equivalent formulations of the axiom of choice. Because it is equivalent to Zorn's Lemma and the well-ordering theorem, you could say it has different names. I could be wrong, but I do not think people knew immediately these things amounted to the same thing. But the equivalences are now considered a part of classical mathematics.

Answer 2

Sidon sets and Golomb rulers. In a Sidon set no two pairs of numbers have the same sum. In a Golomb ruler no two pairs have the same difference. Once you think about it, it's straightforward to see that these conditions are equivalent.

Apostolos Dimitromanolakis's diploma thesis claims:

Although from the nature of these two problems it is apparent that they are related, they have been studied for the most part independently. Consequently, important theoretical results from Sidon set theory were never applied to Golomb rulers.