Introduction to topology

Topology is a natural part of geometry.

As some geometries (such as the spherical geometry) have no good global coordinates system, the existence of coordinates system is put as a local requirement : the existence of correspondences of its small enough regions with those of ℝn. This axiom defined on the weakest kind of geometric structure that is topology, qualifies the given topological space as an n-dimensional manifold.

Topology is one of the weakest kind of structure present in all usual geometrical spaces, including projective and inversive spaces. For now, let us only give intuitive descriptions rather than formal (mathematical) ones, since the possible formalisms will be a bit harder to grasp.

Different possible versions of such a structure can be distinguished in a given space, from the weakest (with "manier" automorphisms), to stronger ones (with "fewer" automorphisms).
Manifolds are kinds of "spaces" where we can distinguish curves, surfaces and other "subspaces" (sub-manifolds) with any dimension, where we may specify diverse degrees of regularity, but no measure of distance or straightness.
There is a range of such possible kinds of "spaces":
There is a standard formalization of topology that works rather well, but it does not look intuitive, is not directly useful for physics, and it does not appear as a particular case that can be directly modified to similarly formalize the diverse further classes of manifolds. Also, it allows for "too many possibilities" that are not physically meaningful.

The weakest one, that is the one with the official name of topology, can intuitively be described as the structure of "connectedness". The isomorphisms for this structure are called homeomorphisms : they are continuous deformations, that include stretching and bending, but not tearing (cutting) or gluing.
Two topological spaces are said to be homeomorphic if there exists an homeomorphism between them.

But this structure also admits morphisms without any requirement of injectivity or surjectivity, which are called the continuous functions. This forms a concrete category which is not a category of relational systems as we shall explain later.

A stronger structure is what we mentioned by qualifying spaces as "smooth" : it is the "approximately affine" structure that is given to every "small" region.
Note that it presumes a notion of what is a "small" region. This notion (which is not really a mathematical notion, but...) is itself a topological notion (in the sense of the weakest topological structure, of connectedness).
The functions serving as morphisms for this structure are called smooth or differentiable (and are particular cases of continuous functions); isomorphisms for this structure are called diffeomorphisms.

In smooth spaces, we have the relation "C and C' are tangent at A" defined between curves C and C' that meet at a point A ; but this relation is not defined in mere topological spaces.
For example, brownian functions are continuous but not smooth. Let us say that the graphs of continuous functions are curves (if a graph of function is a curve then this function is continous, but some curves such as the Koch snowflake, can nowhere be seen as the graph of any function), while graphs of smooth functions are smooth curves.

Diffeomorphisms of the plane are permutations that send straight lines or other smooth curves into other smooth curves, but homeomorphisms can send them into curves that are not smooth, such as graphs of brownian functions.

The dimension of a geometrical space can be defined from its mere topological structure; and it is the only local property of this structure. In other words, two topological spaces (for example, a projective space and an inversive space) have the same dimension if and only if they are locally homeomorphic to each other. Precisely, any "not too big" region of the one, for example a ball-like region, is homeomorphic to such a region of the other (some neighborhood of any given point).
Thus, for a given dimension, projective spaces and inversive spaces are locally homeomorphic. But they are not (globally) homeomorphic, because they do not have the same set of points "near infinity" (and any bijection between these sets would be discontinuous).

Some other topological spaces do not have any definite dimension, as they are not homeomorphic to any ordinary geometrical space.

But the dimension cannot be defined for a mere set of points without the help of topology, as any spaces with different dimensions (except the 0-dimensional ones) are usually in (discontinuous) bijection with each other.

By the action of the automorphism group, the space of points usually has only one orbit (but some have more), so that the group of automorphisms is a manifold with even higher dimension (the dimension of a point's orbit + that of the subgroup of the automorphisms leaving this point fixed). The automorphism group for n2+n)-dimensional affine spaces has dimension (n2+n).
This happens for many finite-dimensional spaces of interest, but yet not all:

The need of second-order structures to express topology

In the Galois connection (Aut, sInv) between structures and permutations, for every set GB, its closure is the automorphism group Aut(sInv G)(E) ⊃ G (the set of all permutations strongly preserving all strong invariants of G). It includes the group G' generated by G, but differs from G' .

Precisely, Aut(sInv G)(E) is the group of permutations that coincide on any n-tuple with some element of the group G' algebraically generated by G, for every number n that we accepted as an arity of a structure : denoting sInv(n)(G) = {rEn | G×{r} ⊂ I(n)},
f∈Aut(sInv(n)G)(E) ⇔ ∀xEn, ∃ gG',  fx = gx.
Connected spaces with dimension >1 do not have any non-trivial first-order structure (finitary relation) invariant by diffeomorphism. Thus, in such spaces, neither topology nor smoothness can be expressed by any first-order structure.

Consequently, the formalization of topology or smoothness for spaces with dimension >1 must consist in one or more second-order structure(s). The notions of curve and smooth curve, are examples of second-order structures, respectively for topology and smoothness.
However, geometries of any dimension can be expressed by mere first-order structures on points, as topology and smoothness need not explicitly appear among the fundamental structures, but may instead merely be definable from them.

Dual spaces can define some more rigid structures than would be possible by the first method. For example, morphisms so defined between infinite dimensional topological affine spaces are automatically "continuous", in a sense of "continuity" that is specific to infinite-dimensional spaces, a condition which the algebraic definition by barycenters does not ensure (while both concepts of morphism are equivalent in finite dimensional spaces).

It is very simple to introduce the notion of measure on a topological manifold M : take M* the vector space of continuous functions with real values, then the space of measures on M is the vector space of linear forms on M* that is "generated by M", i.e. the set of limits of sequences of linear combinations of elements of M in the dual of M*. Now taking as M a differential manifold and M* the set of smooth functions on M, then what we get in this construction (closed vector space generated by M) is the space of distributions on M.

[This page below is a mere bad draft, to be reworked later]

About 1-dimensional topology

In 1-dimensional spaces, topology (but not smoothness) can be expressed by a finitary relation, with arity from 2 to 4 depending on the case.
(Let us forget about non-connected spaces, which are but a union of their connected components so that we only need to put a separate structure on each of them)

A connected 1-dimensional space can either be an "open line" (like a 1-dimensional affine space) or a "loop" (like a circle, or a 1-dimensional projective space that is isomorphic to a straight line in the projective plane). Also, it may be oriented or not. The interest of orientation is that it is not expressed as a separate structure but as a simplification of the form of the topological structure.

The case of 1-dimensional continuous spaces is interesting, as (unlike spaces with dimension 2 or more) it can be rigorously formalized in standard set theory with a first-order structure. However its axiomatization is second-order.

Namely, this structure is that of a total order : a transitive relation < where for any two point A,B we have either A<B or B<A, and ((A<B and B<A
Rather, a structure of total order defines the topology of an oriented line, that is a 1-dimensional space not looped to itself ; we have to reverse the order for the case of the opposite orientation.
We can formalize the same topology without orientation, by using the relation between 3 points (A,B,C) that says "B is between A and C" but this just brings complications without any special interest as the formalization is essentially the same.
As for oriented loops, cutting them at any point would give a line, so that they can also be reduced to the case of an ordered space by adding one variable. And non-oriented loops add up both needs of further variables, thus can be formalized by a relation between 4 points, which is again a further complication without special interest.

So to simplify things we shall stick to the case of the oriented line, formalized by a total order.

Other properties

In a topological space, the measurement of volumes cannot be defined as an operation on tuples, but only on more general figures : it is a second-order structure that cannot be reduced to a first-order one.

But then, once intuitively described the transformations (and understood that they "preserve something" and form a group), what can the topological structure formally consist of ?
There is a standard formal definition of this structure used by mathematicians, but is it a second-order structure: a topology on a set X is a set τ of subsets of X satisfying some properties. (These properties are second-order ones, involving P(τ)).
This formalization is quite unintuitive at first, and requires some time of use and training to be mastered. Moreover, it is of no direct use in physics.

Now we are going to present 2 alternative possible formalizations of topology.

With nonstandard analysis

Let us start with the simplest and most intuitive, but it is a cheat : it rigorously makes no senses in standard set theory, but requires another theoretical framework : to take a nonstandard model of set theory, containing nonstandard numbers ("infinitely big" and "infinitely small" numbers).

So it is a "simple" formalization of topology which can only make sense inside a very special and strange framework which will be specified below. To start with seemingly simple things, here is the definition, which simply looks like a first-order structure: 

The topological structure is the relation between 2 points A and B, expressed as "A and B are infinitely close to each other".

An advantage of this formalization is that we can as simply define from it not just the isomorphisms (deformations), but also the morphisms (the continuous functions, by referring to the general definition of "morphism" for a binary relation:

A continuous function between two topological spaces X and Y is a function f from X to Y so that for any 2 points A and B infinitely close to each other, f(A) and f(B) are also infinitely close to each other.

So, while continuous functions cannot cut, they may smash and glue spaces.

To specify the necessary framework:

What we need is a nonstandard model of set theory. As any model of set theory, it contains its own model of arithmetic which satisfy the second-order axioms of arithmetic as interpreted in this model of set theory. However, as this model does not contain all subsets of its sets, so that "from the outside", it turns out to be a nonstandard model of arithmetic, Its non-standard numbers appear as standard when viewed from the inside of this model of set theory. So we must articulating 2 different views of things : the internal view that does not distinguish which numbers are standard and which are not ; and the "meta" view that makes this distinction. Moreover we must be careful that the set of standard numbers, as a subset of the set of "all" (standard and nonstandard) numbers, only "exists" at the meta level, but does not exist in the internal view.

(to be continued)

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