Many say that Grothendieck has revolutionized the concept of space, but few explain in sufficient detail what this revolution is. I would like to know your position in this regard.
In Grothendieck's philosophy a space is a set together with a collection of coverings that allow you to go from "local" information to global (i.e. way to do glueing or descent). The general notion is that of topos. May be even more important is that he stressed the importance of looking at structure preserving maps between "spaces", as more fundamental and more general than "spaces: themselves, i.e. the categorical point of view. A natural outgrowth of that is the general philosophy that the most useful constructions and invariants are functioral, i.e. respect maps. For example it is often fruitful to think of a space X as the map \pi: X --> point and an invariant of the space e.g. its cohomology H^i(X), as a special case of a pushforward R^i\pi_* associated to \pi. This thinking then naturally leads to greatly generalized and made much more useful general construction for a pushforward R^if_* of a map f: X--> Y. This then leads to study how these pushforwards behave under compositions X--> Y--> Z. Which then subsumes many classic results on cohomology but is also messy and led to a (characteristically abstract) further generalisation (derived categories) where the composition of "pushforwards" is functorial.
The best known example of this philosophy is Grothendieck's construction of schemes. The scheme of a commutative ring spec(R) is the set of prime ideals. It comes with a toplogy (the Zariski topology) and every element r \in R defines a "function" on spec(R). schemes of rings can be glued to more general schemes and in this way he could in a very natural way tie together (and use our geometric intuition!) for things like a one dimensional space of primes of the integers and classic solution sets of algebraic equations.
A nice exposition is Cartier's:
http://library.msri.org/books/sga/from_grothendieck.pdf
which also contains some interesting biographical remarks on Grothendiecks turbulent life.
Dear Mohamed, thanks but I'm looking for a rigorous way to explain the concept of space (in the sense of Grothendieck) for students of the early years of the university.
@Giancarlo, sorry what is your question?
Anyway, I forgot to mention that for physicists perhaps the simplest approach to Grothendiecks idea of schemes is to start with the Gelfand transform as a functor from commutative C^* algebras to locally compact spaces.
Dear Rogier, you say: "In Grothendieck's philosophy a space is a set together with a collection of coverings that allow you to go from "local" information to global."
I assume you mean a small category with a Grothendieck topology, ie a site . But this is not a space, rather it is a presentation of a space.
@Giancarlo, I know I should say small category, to do things like the FPPF (Fidèlement Plat de Présentation Finie) site, and I know, sheaves on a site are just a presentation of the notion of topos, but I was trying to get ideas across. Take the don't stare on one "space", but consider the maps between them as "spaces" as more fundamental as a simple minded (and hopefully easier to understand) way to say that.
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