What are some examples of understanding a space by studying the functions on this space? In Quantum theory, groups and representations, Peter Woit writes:

A fundamental principle of modern mathematics is that the way to
understand a space $M$, given as some set of points, is to look at $F(M)$,
the set of functions on this space.

I was wondering what some examples of this "fundamental principle" is across different fields in mathematics.
Woit, Peter, Quantum theory, groups and representations. An introduction, Cham: Springer (ISBN 978-3-319-64610-7/hbk; 978-3-319-64612-1/ebook). xxii, 668 p. (2017). ZBL1454.81004.
 A: The idea of studying the relationship between structured spaces and appropriate spaces of functions thereon could be described as one of the basic principles of functional analysis, perhaps even the defining one.
Examples:

*

*completely regular spaces and continuous functions—general, bounded or of compact support (in the locally compact case);


*$\sigma$-algebras and (bounded) measurable functions;


*measure spaces and $L^p$-spaces (strictly speaking, equivalence classes of functions);


*smooth manifolds, including open subsets of euclidean space, and spaces of smooth functions, sometimes combined with growth conditions;
and finally, but the list could go on,

*

*complex manifolds and holomorphic functions, again often combined with growth conditions.

The next link in the chain is a consideration of the duals of these function spaces. Here there are two main streams:

*

*representation theorems—the cases where these duals have explicit descriptions, either as spaces of functions themselves or of measures (duality for $L^p$-spaces, Riesz representation theorem);


*the cases where they are used to define new types of objects (Schwartzian distributions, the Bourbakian approach to measure theory).
A: The idea goes back to the 1930s when algebraic geometers understood that points
of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions
on it. The counterpart of this in analysis is the theory of commutative Banach algebras of Gelfand. Later Grothendieck revised  the foundations of the whole algebraic geometry based on this idea, and it spread to many other areas of mathematics.
To answer the comment: van der Waerden's Moderne Algebra, 1-st edition was published in 1930. In it a point (of a Riemann surface) is defined as a certain subring of the field (of meromorphic functions on this Riemann surface). And surely, van der Waerden is not the author of this idea: his book is based on lectures of Artin and Noether.
A: Morse theory is an example of such method. The classification of compact surfaces using Morse theory is done for example in the book of Hirsch, Differential topology. The book of Milnor, Lectures on the h-cobordism theorem goes a step further by proving the Poincare conjecture in dimension bigger than five using Morse theory.
It should be emphasized however that "look at $F(M)$, the set of functions on $M$" is just one way to understand a space $M$ amongst many others.
A: The (finite covering) dimension of $T_{3\frac12}$-topological spaces (completely regular spaces) is characterized by the universal maps into respective cubes.
(Universal maps are not just "into" but always surjective, i.e. onto -- but that's not enough)
