4.4 Topology

Building on the available definitions, additional structures can be introduced, which can be described as being the bones of geometry as well as being useful for programming abstractions.

Definition 28 (Topology) A topology 𝒯 of a set 𝒳 is defined by the following properties:


Both ∅∈𝒯 and 𝒳 ∈  𝒯 .


A finite intersection of members of 𝒯 is in 𝒯 .


An arbitrary union of members of 𝒯 is in 𝒯 .

The concept of a topology alone is incomplete, when not applied to a set, thus providing:

Definition 29 (Topological space) The pair of the set 𝒳 and the topology 𝒯 are denoted as the topological space (𝒳 ,𝒯 )  .

Topological spaces also allow to further qualify the concepts of mappings, as provided by Definition 20.

Definition 30 (Continuous mapping) A mapping between two topological spaces

ϕ : (𝒳 ,𝒯X ) → (𝒴, 𝒯Y )                           (4.33)
is continuous, if the preimages of open sets are open sets.

Equipped with the concept of continuity it is beneficial to specially label a certain class of continuous, invertible mappings.

Definition 31 (Homeomorphism) A continuous mapping ϕ is called a homeomorphism or topological isomorphism, if it satisfies the following properties:

Beside continuity (Definition 30), relations between two mappings from a topological space to another are of interest. Special requirements in this regard are encountered for example in Definition 39, thus motivating the following definition.

Definition 32 (Homotopy) Given two continuous mappings f : 𝒜 →  ℬ and g : 𝒜 →  ℬ , a homotopy h is a continuous mapping such that:

    h : 𝒜 × [0;1 ] → ℬ such that                      (4.34a)

𝒜 × {0 } : h(x,0) = f(x )                           (4.34b)
𝒜 × {1 } : h(x,1) = g(x)                            (4.34c)

A homotopy thusly describes the smooth deformation of one mapping into another.

Further refinement of the concept of a topological space (Definition 29) by including separability of neighbourhoods of different elements of the topological space allows to move closer to the desired settings of geometry described in Definition 37.

Definition 33 (Hausdorff spaces) The topological space (𝒳 ,𝒯 )  is called Hausdorff, if

∀x,y(x ⁄= y ) ∈ 𝒳                              (4.35)
there exist open sets U1, U2  such that x ∈ U1,y ∈ U2  and    ⋂
U1   U2 =  ∅ .

From the concept of a Hausdorff space an important definition can be derived, which is invaluable for use in the discrete settings of digital computers:

Definition 34 (CW Complex) The pair of a Hausdorff space X  (Definition 33) together with a decomposition E , whose elements e ∈ E are called cells, is a CW complex, if the following criteria are met:

A further intermediate definition is introduced, before arriving at the topological space, which carries sufficient additional structure to accommodate the geometric entities (e.g., Definition 54). It serves to again partition the requirements into manageable components, so as not to end up with overly complex and contorted constructs.

Definition 35 (Topological manifold) A topological manifold 𝔐  of dimension n is a Hausdorff space which is locally homeomorphic to ℝn  . Consequently, this results in the existence of bijective mappings κ , called charts, of the form

κ : 𝒰 →  ℳ  ⊆ ℝ  ,                              (4 .36)
which map open neighbourhoods 𝒰 for every element p ∈ 𝔐  to open subsets        n
ℳ  ∈ ℝ  . The n -tuple of numbers resulting from κ (p )  is called the (local) coordinates of p .

As the subsets 𝒰 are open and the charts operate solely on them, any conclusion from a single chart alone must remain a local one. However, open subsets may be combined to overcome this limitation thus giving rise to:

Definition 36 (Atlas) A collection of charts forming an open cover (Definition 1)

𝒜 =    (𝒰i,κi)                               (4.37)
is called an atlas 𝒜 .

Since the atlas is now defined on the whole manifold, it is possible to judge global properties. Considering two charts of an atlas κ (𝒰 ) ∈ 𝒜
 i  i and κ (𝒰 ) ∈ 𝒜
 j  j with the non-empty intersections of open sets

𝒰i ∩ 𝒰j ⁄= ∅                                 (4.38)
it is possible to define chart transitions (a change of coordinates) between these charts within this intersection:
κi ∘ κ−j1 : κj(𝒰i ∩ 𝒰j) → κi(𝒰i ∩ 𝒰j).                   (4.39)
An illustration is given in Figure 4.1.


Figure 4.1: Chart transitions on a manifold.

The following definition finally introduces a topological space, which is equipped to admit the differential structures required in the further geometrical considerations (Definition 48).

Definition 37 (k-Differentiable manifold) A topological manifold 𝔐  with an atlas 𝒜 is called a differentiable manifold, if the transitions between charts ∀κ ∈ 𝒜 are of differentiability class k
C .

The requirement of differentiable transitions between mappings relies on the charts themselves being homeomorphisms (Definition 31) as continuity is a prerequisite for differentiability. The additional structure of a differentiable manifold also allows to place a more stringent requirement on a homeomorphism, which is expressed in the following definition.

Definition 38 (Diffeomorphism) A homeomorphism (Definition 31), which is itself differentiable as well as its inverse, is called a diffeomorphism.

Differentiable manifolds and diffeomorphisms are of particular importance for physical applications, as already attributed by the ancient adage “natura non facit saltus”.