We are now concerned with more radical possibilities.

A paradigmatic example is topology.

In modern “analytic topology”, a “space” is defined to be a set of points equipped with a collection of subsets called open,
which describe how the points vary continuously into each other.
(Most analytic topologists, being unaware of synthetic topology, would call their subject simply “topology.”)

By contrast, in synthetic topology we postulate instead an axiomatic theory, on the same ontological level as ZFC,
whose basic objects are spaces rather than sets.

Of course, by saying that the basic objects “are” spaces we do not mean that they are sets equipped with open subsets.

Instead we mean that “space” is an undefined word,
and the rules of the theory cause these “spaces” to behave more or less like we expect spaces to behave.

In particular, synthetic spaces have open subsets (or, more accurately, open subspaces),
but they are not defined by specifying a set together with a collection of open subsets.

It turns out that synthetic topology, like synthetic set theory (ZFC), is rich enough to encode all of mathematics.

There is one trivial sense in which this is true:
among all analytic spaces we find the subclass of indiscrete ones,
in which the only open subsets are the empty set and the whole space.

A notion of “indiscrete space” can also be defined in synthetic topology,
and the collection of such spaces forms a universe of ETCS-like sets
(we’ll come back to these in later installments).

Thus we could use them to encode mathematics, entirely ignoring the rest of the synthetic theory of spaces.
(The same could be said about the discrete spaces,
in which every subset is open;
but these are harder (though not impossible) to define and work with synthetically.

The relation between the discrete and indiscrete spaces,
and how they sit inside the synthetic theory of spaces,
is central to the synthetic theory of cohesion,
which I believe David is going to mention in his chapter about the philosophy of geometry.)

However, a less boring approach is to construct the objects of mathematics directly as spaces.

How does this work?
It turns out that the basic constructions on sets that we use to build (say) the set of real numbers have close analogues that act on spaces.

Thus, in synthetic topology we can use these constructions to build the space of real numbers directly.

If our system of synthetic topology is set up well,
then the resulting space will behave like the analytic space of real numbers
(the one that is defined by first constructing the mere set of real numbers and then equipping it with the unions of open intervals as its topology).

The next question is,
why would we want to do mathematics this way?

There are a lot of reasons,
but right now I believe they can be classified into three sorts:
modularity,
philosophy, and
pragmatism.

(If you can think of other reasons that I’m forgetting, please mention them in the comments!)

By “#modularity” I mean the same thing as does a programmer:

even if we believe that spaces are ultimately built analytically out of sets,
it is often useful to isolate their fundamental properties and work with those abstractly.

One advantage of this is #generality.
For instance, any theorem proven in Euclid’s “neutral geometry”
(i.e. without using the parallel postulate)
is true not only in the model of ordered pairs of real numbers,
but also in the various non-Euclidean geometries.

Similarly, a theorem proven in synthetic topology may be true not only about ordinary topological spaces,
but also about other variant theories such as topological sheaves, smooth spaces, etc.

As always in mathematics, if we state only the assumptions we need, our theorems become more general.