Topology

6.1 Topological Spaces

In the late nineteenth and early twentieth century the investigation of continuity led to the creation of topology, ¹ a major new branch of mathematics conferring on the idea of the continuous a vast generality. The origins of topology lie both in Cantor ’s theory of sets of points as well as the idea, which had first emerged in the calculus of variations , of treating functions as points of a space.² Central to topology is the concept of topological space. A topological space is a domain equipped with sufficient structure to enable functions between such spaces to be identified as continuous. Now intuitively, a continuous function is one with no “jumps”, that is, it always sends “neighbouring” points of its domain to “neighbouring” points of its range. In order to support the idea of a continuous function in this intuitive sense, the concept of topological space must accordingly embody some notion of neighbourhood . It was in terms of the neighbourhood concept that Felix Hausdorff (1868–1942) first introduced, in 1914, the concept of topological space.

Suppose we are given a set X of elements which may be such entities as points in n-dimensional space, plane or space curves, real or complex numbers, although we make no assumptions as to their exact nature. The members of X will be called points. Now suppose in addition that corresponding to each point x of X we are given a collection ../images/474107_1_En_6_Chapter/474107_1_En_6_Figa_HTML.gif

../images/474107_1_En_6_Chapter/474107_1_En_6_Figa_HTML.gif

of subsets of X, the members of which, called basic neighbourhoods of x are subject to the following conditions:

(i)
each member of contains x;
(ii)
the intersection of any pair of members of includes a member of ;
(iii)
for any U ∈ and any y ∈ U, there is V ∈ such that V ⊆ U.

We may think of each U ∈ N _x as determining a notion of proximity to x: the points of U are accordingly said to be U-close to x. Using this terminology, (i) may be construed as saying that, for any basic neighbourhood U of x, x is U-close to itself; (ii) that, for any basic neighbourhoods U and V of x, there is a basic neighbourhood W of x such that any point W-close to x is both U-close and V-close to x; and (iii) that, for any basic neighbourhood V of x, any point which is U-close to x itself has a basic neighbourhood all of whose points are U-close to x. A subset of S which includes a basic neighbourhood of x is called a neighbourhood of x.

A topological space, in brief, a space, may now be defined to be a set X together with an assignment, to each point x of X, of a collection ../images/474107_1_En_6_Chapter/474107_1_En_6_Figa_HTML.gif of subsets of X satisfying conditions (i)–(iii). As examples of topological spaces we have: the real line ℝ with consisting of all open intervals with rational radii centred on x; the Euclidean plane with consisting of all open discs with rational radii centred on x; and, in general, n-dimensional Euclidean space with ../images/474107_1_En_6_Chapter/474107_1_En_6_Figa_HTML.gif consisting of all open n-spheres with rational radii centred on x. Any subset A of any of these spaces becomes a topological space by taking as neighbourhoods the intersections with A of the neighbourhoods in the containing space.

Hausdorff’s original definition of topological space included the condition:

(iv)
if x ≠ y, then there are members U of and V of such that U ∩ V = ∅.

A topological space satisfying this condition is called a Hausdorff space ³; all the above mentioned spaces are Hausdorff spaces, but there are many examples of non-Hausdorff spaces.⁴

Most topological notions can be defined entirely in terms of neighbourhoods . Thus a limit point of a set of points in a topological space is a point each of whose neighbourhoods contains points of the set: a limit point of a set is then a point which, while not necessarily in the set, is nevertheless “arbitrarily close” to it (lying on its boundary , for instance). The boundary of a subset A of a topological space X is the collection of points x ∈ X which are limit points of both A and its complement X – A. A set is open if it includes a neighbourhood of each of its points and closed if it contains all its limit points: it is easily shown that the closed sets are precisely the complements of open sets.

Topological spaces can also be defined in terms of open sets. Thus we define a topology on a set X to be a family ../images/474107_1_En_6_Chapter/474107_1_En_6_Figc_HTML.gif

../images/474107_1_En_6_Chapter/474107_1_En_6_Figc_HTML.gif

of subsets of X satisfying the following conditions:

(a)
the union of any subfamily of belongs to ;
(b)
the intersection of any pair of members of belongs to ;
(c)
X ∈ and Ø ∈ .

Equipped with a topology ../images/474107_1_En_6_Chapter/474107_1_En_6_Figc_HTML.gif

, X is called a topological space (in this second sense) or simply a space; the members of T are called T -open, open in X, or simply open, sets. Given such a space X, we define a basic neighbourhood of a point x to be an open subset U containing x. It is then readily shown that the families ../images/474107_1_En_6_Chapter/474107_1_En_6_Figa_HTML.gif

of basic neighbourhoods, in this sense, of points x of X satisfy the conditions (i), (ii), (iii) originally laid down for basic neighbourhoods.

A base for a topological space X is a family ../images/474107_1_En_6_Chapter/474107_1_En_6_Figd_HTML.gif of open sets in X such that every open set in X is a union of members of . For example, the family of all open intervals with rational endpoints is a base for ℝ, which accordingly has a countable base.

A concept of discreteness can be introduced for topological spaces: thus a space is discrete if each point in it (more exactly each singleton) is an open set. In a discrete spaces points possess the maximum degree of separation, and the space itself possesses the minimum degree of cohesion.

In a topological space, the set-theoretical complement of an open set is closed but not usually open, so within the topology the “negation” of an open set is the interior of—the largest open set included in—the complement. This implies that the “double negation” of an open set U is not in general equal to U. It follows that, as observed first by Stone and Tarski in the 1930s, the algebra of open sets is not Boolean or classical, but instead obeys the rules of intuitionistic logic . In acknowledgment that these rules were first formulated by A. Heyting , such an algebra is called a Heyting algebra .

The notion of continuous function , or map , between topological spaces, can be defined both in terms of neighbourhoods and open sets. A function f: X → Y between two topological spaces X and Y is said to be continuous, if, for any point x in X, and any neighbourhood V of the image f(x) of x, a neighbourhood U of x can be found whose image under f is included in V, in other words, such that the images of any point U-close to x is V-close to f(x). In the case when X and Y are both the real line ℝ, this condition translates into Weierstrass’s criterion for continuity: for any x and for any ε > 0, there is δ > 0 such that |f(x) – f(y)| < ε whenever |x – y| < δ. In terms of open sets, a function f: X → Y is continuous if and only if, for any open set U in Y, the inverse image f ⁻¹[U] is open in X.

A topological equivalence or homeomorphism between topological spaces is a biunique function which is continuous in both directions, that is, both it and its inverse are continuous. Two spaces are homeomorphic if there exists a homeomorphism between them: intuitively, this means that each space can be continuously and reversibly deformed into the other. A property of a space is topological if, when possessed by a given space, it is also possessed by all spaces homeomorphic to the given one. Topology may now be broadly defined as the study of topological properties.⁵ If we consider geometric figures such as triangles and circles as spaces, we see immediately that, in general, geometric properties such as, e.g., being a triangle or a straight line are not topological, since a triangle is evidently homeomorphic to any simple closed curve and a straight line to any open curve. (To see this, imagine both triangle and line made from cooked pasta or modelling clay.) Topological properties are grosser than geometric ones, since they must stand up under arbitrary continuous deformations.⁶ So, for example, a topological property of the triangle is not its triangularity but the property of dividing the plane into two—“inside” or “outside”— regions, as well as the property that, if two points are removed, it falls into two pieces, while if only a single point is removed, one piece remains. As another example we may consider the properties of one-sidedness or two-sidedness of a surface. The standard one-sided surface—the so-called Möbius strip (or band), discovered independently in 1858 by A. F. Möbius (1790–1868) and J. B. Listing ⁷ (1806–1882)—may be constructed by gluing together the two ends of a strip of paper after giving one of the ends a half twist. Both one- and two-sidedness are topological properties.

Metric spaces constitute an important class of topological spaces. A metric on a set X is a function d which assigns to each pair (x, y) of elements of X a nonnegative real number d(x, y) in such a way that:

$d\left(x,y\right)=d\left(y,x\right),\kern0.75em d\left(x,y\right)+d\left(y,z\right)\ge \ge d\left(x,z\right),\kern0.75em d\left(x,y\right)=0\kern0.24em \mathrm{if}\ \mathrm{and}\ \mathrm{only}\ \mathrm{if}\kern0.24em x=y.$

We think of d(x, y) as the distance between x and y. The first of these conditions then expresses the symmetry of distance, and the second—the triangle inequality—corresponds to the familiar fact that the sum of the lengths of two sides of a triangle is never exceeded by that of the remaining side. A set equipped with a metric is called a metric space . Each metric space may be considered a topological space in which a basic neighbourhood of a point x is the subset of X consisting of all points y at distance < r, for positive r. All of the examples of topological spaces given above are metric spaces , but not every topological space is a metric space. Metric spaces are the topological generalizations of Euclidean spaces.

One of the most important properties a subset of a topological space may possess is that of compactness. To define this concept, we introduce the notion of an open covering of a subset A of a space X: this is defined to be a family of open subsets of X whose union contains A. Now A is compact if every open covering of A has a finite subfamily which is also an open covering of A. Clearly any finite subset of a topological space is compact; compactness may be seen as a topological version of finiteness, or, more generally, boundedness. It can in fact be shown that the compact subsets of any Euclidean space are precisely the closed bounded subsets.

There are a number of different ways of describing connectedness, or cohesion in topological terms. The standard characterization is to call a subset of a space connected if no matter how it is split into two disjoint sets, at least one of these contains limit points of the other. For example, any interval on the real line, and any Euclidean space, is connected in this sense. It is easy to prove that a space is connected if and only if it is not the union of two disjoint nonempty open (or. equivalently, closed) sets.

Another characterization of cohesion derives from Cantor’s original definition of connectedness for point sets in Euclidean spaces. Given two points a and b of a space X, a simple chain from a to b is a collection {A ₁,…, A _n} of subsets of B such that A ₁ (and only A ₁) contains a, A _n (and only A _n) contains b, and A _i ∩ A _j is nonempty if and only if i and j differ by at most 1. Thus each link of the chain intersects just its immediate predecessor and successor (as well as itself), as in Fig. 6.1. A space X is then said to be consolidated if for any pair of points a, b and any open covering ../images/474107_1_En_6_Chapter/474107_1_En_6_Fige_HTML.gif

../images/474107_1_En_6_Chapter/474107_1_En_6_Fige_HTML.gif

of X,

contains a simple chain from a to b. It can then be shown⁸ that any connected space is consolidated .

../images/474107_1_En_6_Chapter/474107_1_En_6_Fig1_HTML.png — Fig. 6.1
Caption

Arcwise connectedness is another, stronger version of cohesion. We say that a space is arcwise connected if every pair of its points can be joined by an arc—that is, a homeomorphic image of a closed interval—in the space. Every arcwise connected space is connected, but not conversely. For metric spaces imposing certain additional conditions along with connectedness ensure that the space is arcwise connected. A space is said to be locally connected if every neighbourhood of any point contains a connected open neighbourhood of that point. Local connectedness means connectedness “in the small”. Now in any space that is both connected and locally connected, each pair of points can be joined by a simple chain of connected sets; such a simple chain can be regarded as an approximation to an arc. When such a space is also a metric space, and is in addition compact , then these simple chains can be refined into arcs, yielding the conclusion⁹ that any compact, connected, and locally connected metric space ¹⁰ is also arcwise connected.

A (topological) continuum is defined to be a compact connected subset of a topological space. Recalling that Cantor defined a continuum as a perfect connected (in his sense) set of points, it is significant that within any bounded region of a Euclidean space Cantor ’s continua coincide with continua in the topological sense.

The study of topological continua has led to a number of intuitively satisfying results. Let us call two subsets of a topological space separated if neither contains limit points of the other; it is then the case that a set is disconnected if and only if it is the union of two nonempty separated subsets. If X is a connected space, we define a cut point of X to be a point x of X such that X – {x} is disconnected; otherwise x is said to be a non-cut point of X. Thus a cut point of a space is one whose removal disconnects the space. For example, every point of ℝ is a cut point, while the end points of a closed interval are its only non-cut points. On the other hand, neither a circle nor any Euclidean space of dimension ≥ 2 has cut points. It can be shown¹¹ that every continuum with at least two points has at least two non-cut points. Moreover, if a metric continuum M has exactly two non-cut points, it is homeomorphic to a closed interval¹²; and if, for any two point x and y, M – {x, y} is disconnected, then M is homeomorphic to a circle.

In Euclidean spaces there is a natural definition of dimension , namely, the number of coordinates required to identify each point of the space. In a general topological space there is no mention of coordinates and so this definition is not applicable. In 1912 Poincaré formulated the first definition of dimension for a topological space; this was later refined by Brouwer , Paul Urysohn (1898–1924) and Karl Menger (1902–1985). The definition in general use today is formulated in terms of the boundary¹³ of a subset of a topological space. Now the topological dimension of a subset M of a topological space is defined inductively as follows: the empty set is assigned dimension −1; and M is said to be n-dimensional at a point p if n is the least number for which there are arbitrarily small neighbourhoods of p whose boundaries in M all have dimension < n. The set M has topological dimension n if its dimension at all of its points is ≤ n but is equal to n at one point at least.

Dimension thus defined is a topological property. Menger and Urysohn defined a curve as a one-dimensional closed connected set of points, so rendering the property of being a curve a topological property. In 1911 Brouwer proved the important result that n-dimensional Euclidean space has topological dimension n, so establishing once and for all the invariance of dimension of Euclidean space under continuous mappings.

6.2 Manifolds

Although the origins of the concept of a manifold may be traced to Gauss’s investigations of the intrinsic properties of surfaces, it was Riemann who, in the mid -nineteenth century, first explicitly introduced the idea in a general form. Riemann had conceived of an n - dimensional manifold as an abstract space locally resembling n-dimensional Euclidean space in the sense that in the vicinity of each point an n-dimensional coordinate system can be introduced and a distance function defined between points whose coordinates are infinitesimally close .¹⁴ This evolved into the more general conception of a differentiable manifold , that is, a manifold possessing a differentiable structure at each point. It is our purpose here to describe a special type of differentiable manifold—the so-called smooth manifolds .¹⁵

The modern concept of manifold is based on the idea of a chart. Given a topological space T, let U be a nonempty open subspace¹⁶ of T which is homeomorphic to an open subspace X of the n-dimensional Euclidean space ℝ ⁿ . A homeomorphism σ : p ↦ p ^σ ¹⁷ of U with X is called a chart on U (or in T). In a given chart σ on U, each point p of U corresponds to a point x = p ^σ of X, so that p may be identified with (x ₁,…, x _n), the coordinates of x. The real numbers x _i are called the coordinates of p (in the chart σ), and n is the dimension of the chart .

Now suppose that there is a homeomorphism Φ of X with a another open subspace Y of ℝ ⁿ and let Ψ be its inverse. If y = x ^Φ is a general point of Y with coordinates y _i (i = 1,…, n), then Φ and Ψ may be described by means of continuous functions φ_i, ψ_i: R ⁿ → R:

../images/474107_1_En_6_Chapter/474107_1_En_6_Figf_HTML.png

or simply, writing x for (x ₁,…, x _n) and y for (y ₁,…, y _n),

${y}_i={\upvarphi}_i(x)\kern0.75em \left(x\in X\right)\kern1.00em {x}_i={\uppsi}_i(y)\kern0.75em \left(y\in Y\right).$

(6.1)

Notice that the composite Φ _° σ is a homeomorphism of U with Y, and therefore also a chart on U.

Conversely, if σ and τ are any two charts on the same subspace U of T, mapping U into X and Y respectively, then Φ = τ _° σ⁻¹ is a homeomorphism of X with Y having inverse Ψ = σ _° τ⁻¹ so that the coordinates x and y of corresponding points in X and Y are related by equations of the form (6.1). This shows that, if we regard the passage from x to y as a change of coordinates, then the Eq. (6.1)—with continuous φ_i and ψ_i —are the most general equations defining a change of coordinates.

A function from a subspace of ℝ ⁿ to ℝ is said to be smooth if it has partial derivatives of arbitrarily high orders. Two charts in T whose coordinates are related by the Eq. (6.1) are said to be smoothly related at a point p of T, if they are defined on a neighbourhood of p and if the functions φ_i and ψ_i occurring in (6.1) are smooth. If two charts are smoothly related at every point of T at which they are defined, they are said to be smoothly related.

A topological space is said to be locally Euclidean at a point p if there exists a chart σ on a neighbourhood of p. A manifold is a Hausdorff space which is locally Euclidean at each of its points. It follows that in a manifold M each point has a chart defined on some neighbourhood, so that the family of charts in M may be said to cover M.

We now define a smooth structure on a Hausdorff space M to be a family of charts ../images/474107_1_En_6_Chapter/474107_1_En_6_Figg_HTML.gif

in M satisfying the three following conditions:

(i)
At each point of M there is a chart which belongs to ;
(ii)
Any two charts of are smoothly related;
(iii)
Any chart in M which is smoothly related to every chart of itself belongs to .

Conditions (i) and (ii) are naturally expressed by saying that ../images/474107_1_En_6_Chapter/474107_1_En_6_Figg_HTML.gif

is smooth and maximal, respectively. A smooth structure on M is accordingly a maximal smooth family of charts covering M. It is plain that a Hausdorff space with a smooth structure is necessarily a manifold—with this structure the space is called a smooth manifold . Members of the smooth structure are called admissible charts. It can be shown¹⁸ that any smooth covering family of charts on a Hausdorff space is contained in a unique maximal smooth family of charts.

Let M be a manifold and f a real-valued function defined on a subspace (possibly all) of M: we shall express this by saying that f is defined in M. If σ is a chart on some subspace U on which f is defined, the latter determines a function f∗ from a subspace of ℝ ⁿ to ℝ by defining f∗(x) = f(p), where x = p ^σ. If M is a smooth manifold , a real-valued function f in M is said to be smooth at a point p if it is defined on some neighbourhood of p and its expression f∗ in terms of an admissible chart σ is a smooth function . (This definition does not depend on the choice of the chart σ.) A function which is smooth at every point at which it is defined is said to be smooth.

Finally, suppose we are given two smooth manifolds M and N and a map Φ: M → N. For each function f defined in N we define the function f _Φ in M by:

${f}_{\Phi}(p)=f\left({p}^{\Phi}\right)\kern0.5em \left(p\in M\right).$

Then the map Φ: M → N is said to be smooth if f _Φ is a smooth function in M whenever f is a smooth function in N.

Manifolds and topological spaces give rise to categories, the topic of our next chapter.

Footnotes

From Greek topos, “place” . Here we shall be concerned with what has become known as general topology, as opposed to algebraic or combinatorial topology.

The study of abstract spaces , which led to the definition of topological space was initiated by Maurice Fréchet (1878–1903) in 1906.

Thus points in a Hausdorff space are, as it were, “housed off”.

For example, the real line ℝ with basic neighbourhoods at each point x reduced to the single set {y: x ≤ y}.

In view of the fact that a doughnut and a coffee cup (with a handle) are topologically equivalent, John Kelley (in Kelley 1955) famously defined a topologist to be someone who cannot tell the two apart .

That is, topological properties are chosen so as to satisfy Leibniz’s Principle of Continuity.

It was also Listing who, in his work Vorstudien zur Topologie of 1848, first uses the term “topology”; the subject being known prior to this as analysis situs, “positional analysis”.

Hocking and Young (1961), p. 108.

Ibid., p. 116.

A metric space with these properties is called a Peano space or Peano continuum.

Hocking and Young (1961), p. 49.

Ibid., p. 54.

Defined above.

The distance ds between points (x ₁,…, x _n) and points (x ₁ + dx ₁,…, x _n + dx _n) being given by

../images/474107_1_En_6_Chapter/474107_1_En_6_Equb_HTML.png

where the g _ij are functions of the coordinates (x ₁,…, x _n), g _ij = g _ji and the right-hand side is always positive. This expression for ds ² is a generalization of the usual Euclidean distance formula

../images/474107_1_En_6_Chapter/474107_1_En_6_Equc_HTML.png

My account has been adapted from Ch. 1 of Cohn (1957).

If T is a topological space , and A a subset of T, the topology on T induces a natural topology on A—the relative topology—defined by identifying the open subsets of A as precisely the intersections with A of the open subsets of T. The subset A, with the relative topology so defined, is called a subspace of T.

For convenience we write p ^σ for the value σ(p) of σ at p, and similarly below.

Cohn (1957)., Theorem 1.2.1.

6. Topology

6.1 Topological Spaces

6.2 Manifolds