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
ℝ^{n}. 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":
- Topological spaces
- Topological manifolds
- Lipschitz structures on topological manifolds,
- Differential manifolds (with whatever degree of smoothness you
choose).
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
n^{2}+n)-dimensional affine spaces has dimension
(n^{2}+n).
This happens for many finite-dimensional spaces of interest, but
yet not all:
- Some spaces, such as arbitrary Riemannian manifolds (rigid but inhomogeneous spaces), may have
no automorphism except the identity.
- Other ones, such as the pure manifolds (topological spaces, with roughly no further structures), have a "space" of
automorphisms that can best be qualified as
infinite-dimensional.
The need of second-order structures to express topology
In the Galois connection (Aut, sInv) between structures and
permutations, for every set G⊂B, 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)
= {r⊂E^{n} | G×{r} ⊂ I^{(n)}},
f∈Aut_{(sInv(n)G)}(E)
⇔ ∀x∈E^{n}, ∃ g∈G', f০x
= g০x.
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.
Proof:
In such a space, for any 2 tuples of the same number of pairwise
distinct points (i.e. injective tuples of points), (A_{1},
...,A_{n}),
(B_{1},...,B_{n}), there exists a
diffeomorphism f (automorphism for smoothness) such that
(f(A_{1}),...,f(A_{n})) =
(B_{1},...,B_{n}).
This can be obtained as a composite of diffeomorphisms that
progressively move each Ai to a position near Bi (nearer than to
any of the other points in the list), and finally once each point
is near its target it is no more stopped from reaching it by any
other point in the list.
(This proof is intuitive and not rigorous as the word "near" does
not really make sense without using any distance operation, but
formalized topology can express a proof similar to this).
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 oriented topology of an open line, is that of a total
order (with more axioms). So, it is a binary relation.
- The non-oriented topology of an open line, can be expressed
by the 3-ary relation of betweenness.
- The oriented topology of a loop, can be expressed by a 3-ary
relation.
- The non-oriented topology of a loop, can be expressed by a
4-ary relation.
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.
There is a diversity of non-isomorphic linear orders, beyond finite ones and that of R. (while only the case of R comes in usual theories of physics). Examples : the set of rationals ; the set of irrationals ; the long line ; lines where all countable monotone sequences converge but that are incomplete since gaps have uncountable cofinality; Suslin lines, whose existence is undecidable in ZF.
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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