It is a mathematical device to define what a continuous function is, in a general setting.
A topological space is a set X together with a collection of subsets of X, called open, such that finite intersections of open sets are open, and arbitrary unions of open sets are open.
A function of topological spaces is continuous if inverse images of open sets are open.
What is the intuition behind (1) and (2)?
I claim that it is better to ask, instead, how mathematicians came up with (1) and (2).
Mathematicians spoke of continuous functions for a long time before there was a precise definition of continuous function. This was a vague idea, which, nevertheless, was useful.
At some point the definition of continuity for a function f : ℝ → ℝ was elucidated.
This definition allowed a lot of theorems to be proved rigorously. This is why it was useful.
A lot of theorems that were claimed, could now be proved.
2/
Suppose you want to calculate f(x₀), for example because you are an engineer and want to build a bridge, where x₀ is a physical quantity. Unfortunately, we can't measure x₀ exactly.
But you still want to know what y₀ = f (x₀) is, at least approximately.
But "approximately" doesn't mean anything. You want to know y₀, say, with two correct decimal digits. This will do to build a robust bridge.
Then the question is, how many correct digits of x₀ do you need to know, in order to get the desired two decimal digits of y₀?
More generally, we want the following to be the case. In order to know n digits of output of the function, it is enough to know m digits of the input, where m depends on the input and on n.
This is possible if and only if the function is continuous.
So one intuition about continuity is that "finite amounts of output depend on only finite amounts of input".
Soon people started to consider continuous functions "of two variables", that is, functions ℝ × ℝ → ℝ, and then of n variable, so ℝⁿ → ℝ, and in each case a different definition was needed.
Moreover, because people were trying to e.g. solve differential equations, which amounts to given one input function (called the initial condition) to figure out an output function (the solution), people came across functions mapping continuous functions to continuous functions, which, themselves, may or may not be continuous.
So a general definition of continuity was needed to clean up the mess and be able to make progress more efficiently.
A first, rather useful, general setting was that of a metric space. You say what the distance between two things (e.g. real numbers, tuples of real numbers, continuous functions) is. The axioms for metric spaces are very intuitive, and I won't reproduce their statement here.
But I want to say this.
The definition of continuity is the same as above, with the absolute value of the difference replaced by the distance function d. A function f : X → Y is continuous iff
∀ x ∈ X , ∀ ε > o, ∃ δ > 0 , ∀ x' ∈ X, d(x,x') < δ → d(f(x),f(x')) < ε.
We can then define a set U to be open if for every x ∈ U there is ε > 0 such that every x' with d(x,x') < ε is in U.
The open sets are closed under finite intersections and arbitrary unions.
A function is continuous in the above ε-δ sense iff inverse images of open sets are open.
What happens next is that metric spaces are not enough. There are sets on which we want to consider continuous functions which can't be metrized so that metric continuity coincides with the notion of continuity we want.
I like this book because it starts from trivial things, making them less and less trivial as we progress, until it gets eventually to many things, including Stone duality. I self-learned topology from this book as an undergrad.
Then comes computer science. Smyth, Abramsky and Vickers propose a radically different intuition for the axioms of topology and the definition of continuous function.
I will stop this post at this point, at least for now, giving four seminal references.
But one should say the following. Actually, the connection of topology with computation started with Brouwer, who was a topologist himself, in fact one of the founding fathers of topology before Hausdorff come up with the axiomatization of topological spaces which I described above.
And finally, in the same way metric spaces were not enough, and so we moved from metric spaces to topological spaces, Grothendieck found that topological spaces were not enough for him, and so he needed, more generally, toposes.
And recently condensed sets and pyknotic sets have also been proposed as a convenient generalization of the notion of (topological) space.
It is easy to lose track of the original intuitions. It is also easy for new intuitions to replace the original ones, when we move to more general settings. And we always move to more general settings in mathematics.
@MartinEscardo do you know, was Grothendieck's issue essentially that the Zariski topology just doesn't have very many open sets? Or was there something more to it?
@joshuagrochow@MartinEscardo I believe Serre noticed something like the fact that G bundles are not classified by cohomology classes, but they are if you pull back by étale maps instead
@boarders@joshuagrochow@MartinEscardo
Indeed, it was already a surprise when Serre proved that the Zariski topology could recover the sheaf cohomology of coherent sheaves ,(FAC and GAGA papers), but that did not furnish the “correct” cohomology groups that would help proving the Weil conjectures (along the way suggested by Weil himself).
At some point, Serre observed that “algebraic fiber spaces” were not locally trivial for the Zariski topology, but we're made trivial after passing to an étale covering.
He made a talk at the Chevalley seminar about it
Grothendieck was there and, at the end of the talk, told Serre that they had (« on a ») the Weil cohomology.
@antoinechambertloir@boarders@joshuagrochow@MartinEscardo - there is an amusing interplay between the words "cover" and "covering" in mathematics. As I started learning topology I thought an "open cover" was something completely different from a "covering space" - indeed, strangely, it never occurred to me to wonder if these concepts were related. But then, learning concepts like "etale map" and "etale topology", and learning that an open cover (U_\alpha) of a space (X) gives a local homeomorphism (\sum_\alpha U_\alpha \to X), I realized that open covers and covering spaces are deeply related. Of course this is something that all experts already know.... but I became curious why people don't run around saying "hey! coverings and covers both give etale maps!"
I have no idea what Serre and Grothendieck and others thought about "covers versus coverings" - some of this may be special to English-language math - but Serre's observation that "algebraic fiber spaces" can be trivialized by pulling back along an étale covering sounds like exactly the right reason to develop algebraic geometry in a way that emphasizes their relationship.
@johncarlosbaez@boarders@joshuagrochow@MartinEscardo
There is no such phonetic proximity in french, revêtement et recouvrement are quite different words. However, both Serre and Grothendieck had thought a lot about topology, having thought about fiber spaces in particular.
In his PhD thesis, Serre generalized the concept of Hurewicz fibration to what we now call Serre fibration— requiring the lifting property for maps from cubes only.
The first work of Grothendieck in algebraic geometry was in 1955, about fiber spaces.
@antoinechambertloir - It's even more confusing because in English mathematicians say "covering space" (never "cover space") and "open cover" (never "open covering") where the word is being used as an adjective, but "SU(2) double covers SO(3)", where the word is being used as a verb.
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