I. Polyhedra, Manifolds, Topological Spaces
A. We begin with the concept of a simplex. A zero-dimensional simplex is a point; a one dimensional simplex is a straight line segment. A two dimensional simplex is a triangle (including the plane region which it bounds), a three dimensional simplex is a tetrahedron. It is known and easy to show that if one considers all possible distributions of non-negative point masses at the four vertices of a tetrahedron the point set consisting of the centers of mass of these distributions is precisely the tetrahedron itself; this definition extends easily to arbitrary dimensions. We assume here that r+1 vertices of an r dimensional simplex are not contained in an (r-1)-dimensional hyperplane of the we are considering.
Definition of a simplex: One could also define a simplex as a smallest closed convex set which contains the given vertices.
Recall the following definitions:
Closed set: The complement of an open set. (Rudin says any open ball is an open set. He also says that a subset E of a metric space X is closed if every limit point of E is a point of E. Also, a subset E is open if every point of E is an interior point of E. A point p is an interior point of E if there is a neighbourhood N of p such that where X is the metric space under consideration).
Bounded set: Again, Rudin says: E is bounded if there is a real number M and a point such that for all . Note here d is the distance/metric function under consideration.
Convex set: We call a set convex if : whenever , and . (Open balls are convex, closed balls are convex, k-cells are convex).
Open cover: By an open cover of a set E in a metric space X we mean a collection of open subsets of X such that
Compact set: A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. More explicitly, the requirement is that if is an open cover of K, then there are finitely many indices such that
The notion of compactness is of great importance in analysis, especially in connection with continuity.
Connected Sets: Two subsets A and B of a metric space X are said to be separated if both and are empty, that is, if no point of A lies in the closure of B and no point of B lies in the closure of A. A set is said to be connected if E is not a union of two nonempty separated sets.
in geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined.
Any of the vertices of an r-dimensional simplex define an s-dimensional simplex — an s-dimensional face of the given simplex (the zero dimensional faces are the vertices). Then, we mean by an r-dimensional polyhedron, a point set of which can be decomposed into r-dimensional simplexes in such a way that two simplexes of this decomposition either have no points in common or have a common face (of arbitrary dimension) as their intersection. The system of all the simplexes (and their faces) which belong to a simplicial decomposition of a polyhedron is called a geometrical complex.
The dimension of a polyhedron is not only independent of the choice of simplicial decomposition, but indeed it expresses a topological invariant of the polyhedron; that is to say, two polyhedra have the same dimension if they are homeomorphic (if they can be mapped onto another in a one-to-one and bicontinuous fashion).
With this general viewpoint of topology in mind (according to which two figures — that is, two point sets — are considered equivalent if they can be mapped onto one another topologically), we shall understand a general or curved polyhedron to be any point set which is homeomorphic to a polyhedron (defined in the above sense, that is, composed of ordinary “rectilinear” simplexes). Clearly, curved polyhedra admit decomposition into “curved” simplexes (that is, topological images of ordinary simplexes); the system of elements of such a decomposition is again called a geometrical complex. (Comment: Such decompositions are also partitions due to apt equivalence relations. So mutually exhaustive, exclusive, disjoint equivalent partititions).
B. The most important of all polyhedra, indeed, even the most important structures of the whole of general topology, are the so-called closed n-dimensional manifolds . They are characterized by the following two properties. First, the polyhedra must be connected (that is, it must not be composed of several disjoint sub-polyhedra); second, it must be “homogeneously n-dimensional” in the sense that every point p of possesses a neighbourhood which can be mapped onto the n-dimensional cube in a one-to-one and bicontinuous fashion, such that the point p under this mapping corresponds to the center of the cube.
C. In order to recognize the importance of the concept of the manifold, it suffices to remark that most geometrical forms whose points may be defined by n parameters are n-dimensional manifolds; to these structures belong, for example, phase-spaces of dynamical problems. These structures are, to be sure, only rarely defined directly as polyhedra; rather, they appear —- as the examples of phase-spaces or the structures of n-dimensional differential geometry already show — as abstract spatial construction, in which a concept of continuity is defined in one way or another; it turns out here (and it can be proved rigorously under suitable hypotheses) that in the sense of the above mentioned definition of continuity the “abstract” manifold in question can be mapped topologically onto a polyhedron, and thus falls under our definition of manifold. In this way, the projective plane, which is first defined as an abstract two-dimensional manifold, can be mapped topologically onto a polyhedron of four-dimensional space without singularities of self-intersections.
D. Just one step leads from our last remarks to one of the most important and at the same time most general concepts of the whole of modern topology —- the concept of topological space. A topological space is nothing other than a set of arbitrary elements (called “points” of the space) in which a concept of continuity is defined. Now this concept of continuity is based on the existence of relations, which may be defined as local or neighbourhood relations — it is precisely these relations which are preserved in a continuous mapping of one figure onto another. Therefore, in more precise wording, a topological space is a set in which certain subsets are defined and are associated to the points of the space as their neighbourhoods. Depending upon which axioms these neighbourhoods satisfy, one distinguishes between different types of topological spaces. The most important among them are the so-called Hausdorff spaces (in which the neighbourhoods satisfy the four well-known Hausdorff axioms).
These axioms are the following:
(a) To each point x there corresponds at least one neighbourhood ; each neighbourhood contains the point x.
(b) If and are two neighbourhoods of the same point x, then there are exists a neighbourhood , which is a subset of both.
(c) If the point y lies in , there exists a neighbourhood , which is a subset of .
(d) For two distinct points x, y there exists two neighbourhoods without common points.
Using the notion of neighbourhood, the concept of continuity can be immediately introduced: Definition: A mapping f of a topological space R onto a (proper or improper) subset of a topological space Y is called continuous at the point x, if for every neighbourhood of the point one can find a neighbourhood of x such that all points of are mapped into points of by means of f. If f is continuous at every point of R, it is called continuous in R.
E. The concept of topological space is only one link in the chain of abstract space constructions which forms an indispensable part of all modern geometric thought. All of these constructions are based on a common conception of space which amounts to considering one or more system of objects — points, lines, etc. — together with systems of axioms describing the relations between objects. Moreover, this idea of a space depends only on these relations and not on the nature of the respective objects. Perhaps this general standpoint found its most fruitful formulation in David Hilbert’s Grundlagen der Geometrie; however, I would especially emphasize that it is by no means only for investigations of the foundation that this concept is of decisive importance, but for all directions of present-day geometry — the modern construction of projective geometry as well as the concept of a many-dimensional Riemannian manifold (and, indeed, earlier still, the Gaussian intrinsic differential geometry of surfaces) may suffice as examples!
F. With the help of the concept of topological space we have at last found an adequate formulation for the general definition of a manifold: A topological space is called a closed n-dimensional manifold if it is homeomorphic to a connected polyhedron, and furthermore, if its points possess neighbourhoods which are homeomorphic to the interior of the n-dimensional spheres.
G. We will now give some examples of closed manifolds.
The only closed one-dimensional manifold is the circle. The “uniqueness” is of course understood here in the topological sense: every one-dimensional closed manifold is homeomorphic to the circle.
The closed two-dimensional manifolds are the orientable (or two sided) and non-orientable (or one-sided) surfaces. The problem of enumerating their topological types is completely solved.
As examples of higher-dimensional manifolds — in addition to n-dimensional spherical or projective space — the following may be mentioned:
(i) The three-dimensional manifold of line elements lying on a closed surface F. (It can be easily proved that if the surface F is a sphere then the corresponding is projective space). (Please refer to Wikipedia for projective space)
(ii) The four-dimensional manifold of lines of the three dimensional projective space.
(iii) The three-dimensional torus-manifold: it arises if one identifies the diametrically opposite sides of a cube pairwise. You may confirm without difficulty that the same manifold may also be generated if one considers the space between two coaxial torus surfaces (of which one is inside the other) and identifies their corresponding points.
The last example is also an example of the so-called topological product construction — a method by which infinitely many different manifolds can be generated, and which is, moreover, of great theoretical importance. The product construction is a direct generalization of the familiar concept of coordinates. One constructs the product manifold from the two manifolds and as follows: the points of are the pairs , where x is an arbitrary point of and y an arbitrary point . A neighbourhood of the point is defined to be the collection of all points such that x belongs to an arbitrarily chosen neighbourhood of and y belongs to an arbitrarily chosen neighbourhood of . It is natural to consider the two points x and y (of and respectively) as the two “coordinates” of the point of .
Obviously this definition can be generalized without difficulty to the case of the product of three or more manifolds. We can now say that the Euclidean plane (not a closed manifold) is the product of two straight lines, the torus is the product of two circles, and the three-dimensional torus-manifold, the product of a torus-surface with the circle (or the product of three circles). As further examples of manifolds one has, for example, the product of the surface of a sphere with the circle, or the product of two projective planes, and so on. The particular manifold may also be obtained if one considers the spherical shell lying between two concentric spherical surfaces and and identifies the corresponding points (that is, those lying on the same radius) of and . Only slightly more difficult is the proof of the fact that, if one takes two congruent solid tori and (in accordance with the congruence mentioned) idenfities the corresponding points of their surfaces with one another, one likewise obtains the manifold . Finally, one gets the product of the projective plane with the circle if in a solid torus one identifies each pair of diametrically opposite points of every meridian circle.
Ref: Elementary Concepts of Topology by Paul Alexandroff, Dover Publications, available Amazon India.