Do topological spaces deserve their name?

Question

In algebra, the names of structures don't mean anything outside mathematics : monoids, groups, rings, fields, ... This is good, because it disconnects the mathematical study of those structures from their practical applications. Groups don't promise to model symmetries of a physical system ; the fact that they do is a happy accident.

Topological spaces, however, hack many words that do have meanings outside maths : open, closed, neighborhood, continuous, limit, ... This is a lot more daring, as those concepts were studied long before the discovery of topological spaces in the 20th century.

And the way topology redefines those concepts is questionnable. It observes that open sets are stable by arbitrary unions and finite intersections. To my geometrical intuition, those rules are indeed necessary. But then the definition suddenly stops, and declares that those rules are sufficient to describe open sets. This is a bit like observing that Elvis is a man and is a singer, then stopping to declare that Elvis is any man who sings.

I therefore reckon that open sets of a topological space are something more general than the usual geometrical intuition means. This is probably why obvious geometrical theorems like the Jordan curve, or the fact that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^p$, are so hard to prove, when interpreted in the definitions of topological spaces. And also probably why there are wild beasts like continuous surjections from $[0,1]$ to $[0,1]^2$.

Last example : the Zariski topology in algebraic geometry. According to this topology, closed sets of $\mathbb{R}^n$ are nullsets of $n$-variable polynomials. Those sets do comply with the rules of arbitrary intersections and finite unions, but what is the connection between polynomials and geometrical frontiers ? Once again I believe this is a sign that closed sets of topological spaces are something more general than the geometrical intuition means.

All the previous cases are good theorems, hard theorems. The Zariski topology is quite neat in algebraic geometry. But their interpretation is confused by the use of geometrical words that probably simplify too much what topological spaces speak of. I don't have better names to offer, but then must I ? Why didn't topology choose neutral names like monoids or groups ?

Answer 1

You raise some good points, but first note that your objections in paragraph four do not support your claims. The problems you describe are about very ordinary spaces, well-studied before topology was born.

Algebra, by its nature, is much more abstract than geometry. That is probably why you find topology formalizing so many concepts you use in English regularly, while algebra formalizes more obscure words.

Whether or not it is desirable for mathematics to 'take over' the meaning of words is, perhaps, debateable. When done well, the result is good mathematics and that should be the decisive factor. By the way, algebra also takes over words, for instance a definition being natural is, due to category theory, a formal concept.

More specifically to your concerns as to the definition of topology in terms of abstract open sets, do note topology can be defined in other ways, perhaps less objectionable ones. In particular, topology can be defined metrically in the sense explained in this paper.

Answer 2

Interesting "question". It is true that naming a new mathematical object or whole domain (as is the case here) is not a little thing. A good approach for this is to take an historical point of view. It has taken many many years before this domain has been identified. From say 1860 to 1920-1930. for example, Poincaré (died in 1912), who has been one of the foremost mathematician to work on the subject has coined a name "analysis situ" which has not been retained. As he was not in the axiomatic trend led by the other contemporary giant, Hilbert, he has not reached the axiomatic definition we know, set in 1914 by Hausdorff, and then later by Kolmogorov and others.

You may take interest to see the subtle axiomatic development in (https://en.wikipedia.org/wiki/History_of_the_separation_axioms)

Answer 3

I don't think the claims regarding names in algebra are particularly on-point. The word group, for example, came from the longer phrase group of substitutions, which was a synonym for collection of permutations. In many languages (though not in English) the word for a field is actually body, as in body of numbers, similar to body of water, meaning a collection of numbers. In short, many of the names in algebra are just fairly generic collective nouns (similar to the word set as it is used in mathematics). They are not particularly evocative, but they are not completely random either.

As for the remarks about, e.g. the Zariski topology, these are also not really correct, in my view. One could see e.g. this answer for an alternative perspective on the Zariski topology, explaining how to thing about it with genuine topological intuition.