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Cartesian product. Projection function. Product topology. Product space. Subbase and base for product topology. Metric product spaces

Def. Cartesian product A B of two sets A and B. The Cartesian product A B (read “A cross B”) of two sets A and B is defined as the set of all ordered pairs (a, b) where a is a member of A and b is a member of B.

Syn. Product set, direct product, direct sum.

Example 1. If A = {1, 2, 3} and B = {a, b} the Cartesian product A B is given by

A B = { (1, a), (1, b), (2, a), (2, b), (3, a), (3, b) }

Comments. Some comments are in order in regard to the concept of a Cartesian product of two arbitrary sets A and B. One is certainly justified in asking: Does this concept make any sense? What meaning can be attached to a Cartesian product? What meaning can be attached to the ordered pairs? Well, in general, an ordered pair has no meaning unless one has been assigned. In specific cases, when an ordered pair does have meaning, the concept of a Cartesian product becomes meaningful and useful. The Cartesian product finds meaning and use in various places, for example the theory of such abstract mathematical systems as groups, rings, vector spaces and topological spaces. One can view the concept of a Cartesian product as a generalization / abstraction of a concept relating to the Cartesian plane. That concept is this: The set of all points in the Cartesian plane can be viewed as the set of all ordered pairs (x, y) where x ε R and y ε R, R being the set of all real numbers. In fact, a Cartesian product is so defined that R R is the set of all points in what we know as the Cartesian plane. Thus the motivation for the term. In analogy to the way we view number pairs (x, y) as points in the Cartesian plane we can view the ordered pairs of a Cartesian product as points in a Cartesian frame. Figure 1 shows such a representation for the sets A = {1, 2, 3, 4} and B = {a, b, c}.

Example 2. Let A be the set of numbers in the interval [3, 5] and B be the set of numbers in the interval [2, 3]. Then the Cartesian product A B corresponds to the rectangular region shown in Fig. 2. It consists of all points (x, y) within the region. In the same way, if A is the set of numbers in the interval [3, 5], B is the set of numbers in the interval [2, 3] and C is the set of numbers in the interval [6, 7] the Cartesian product A B C

consists of all points (x, y, z) in a rectangular parallelepiped in three-dimensional space defined by

3 x 5

2 y 3

6 z 7 .

Example 3. Let I denote the unit interval [0, 1] and C1 the interior and boundary of the unit circle. Then I I is the unit square, C1 I is a cylinder and C1 C1 is a torus. See Fig. 3.

The Cartesian product A1 A2 An . The Cartesian product A1 A2 An of n sets A1, A2, ...... , An is the set of all ordered n-tuples (a1, a2, .... , an) for ai ε Ai, where each ai assumes all the values in Ai, i = 1, 2, ...., n. We can view an n-tuple (a1, a2, .... , an) as a point in an n-dimensional space defined by the Cartesian product A1 A2 An . The component sets A1, A2, , An of the product are called the coordinate sets of the space. The set Ai is called the i-th coordinate set of the product. The i-th component of the n-tuple (a1, a2, .... , an) is called the i-th coordinate of the vector (a1, a2, .... , an) in this n-dimensional space. This i-th coordinate of the point is called the projection of the vector (a1, a2, .... , an) onto the i-th coordinate set Ai.

The above represents a generalization of concepts associated with the usual 3-space R3 and n-space Rn. The space Rn becomes a special case of the above when A1 = A2 = .... = An = R.

Notations. The Cartesian product A1 A2 Ai of an indexed collection of sets {Ai}i ε I is sometimes denoted by

Projection function. There is a function called the projection function that works as follows: Let X = (a1, a2, .... , an) be a point in the n-dimensional space defined by the Cartesian product A1 A2 An . The projection function πi(X) is defined as

πi(X) = ai

where ai is the i-th coordinate of the point X = (a1, a2, .... , an). Here ai represents the projection of the vector (a1, a2, .... , an) onto the i-th coordinate set Ai , hence the name. The projection function π is a function from the n-dimensional space defined by the Cartesian product A1 A2 An into the i-th coordinate set Xi.

Example 4. Let R1, R2 and R3 denote copies of R. Consider the point X = (3.1, 6.5, 2.8) in three dimensional space R3 = R R R = R1 R2 R3. Then

π1(X) = 3.1 is the projection of X in R1.

π2(X) = 6.5 is the projection of X in R2.

π3(X) = 2.8 is the projection of X in R3.

Cartesian product R R R. The Cartesian product R R R corresponds to the set of all points in three-dimensional space i.e. the set of all number triplets (x, y, z), x ε R, y ε R, z ε R.

Cartesian products R2, R3, .... , Rn. The Cartesian product R R is usually denoted by R2, the Cartesian product R R R is usually denoted by R3 and the Cartesian product R R ..... R (n times) consisting of all n-tuples (x1, x2, .... , xn) is usually denoted by Rn.

Generally a Cartesian product A B is thought of as a two dimensional array of points with each point corresponding to an ordered pair (x, y), a Cartesian product A B C is thought of as a three dimensional array of points with each point corresponding to an ordered triple (x, y, z) and a Cartesian product A1 A2 An is thought of as an n-dimensional array of points with each point corresponding to an n-tuple (or n-vector). An exception to this is illustrated in Example 3 above because C1 is two-dimensional.

Product topology. Product space.

Def. Product topology. Let X and Y be topological spaces. The product topology on the Cartesian product X Y of the spaces is the topology having as base the collection B of all sets of the form U V, where U is an open set of X and V is an open set of Y.

Example 5. Consider the interpretation of this definition for the case when X and Y are R, the set of real numbers. Open sets in R correspond to collections of open intervals. See Fig. 4. U and V are open sets in R and the collection B of all sets of the form U V is a base for the product topology on R R.

Base for product topology

Theorem 1. If B is a base for the topology of X and C is a base for the topology of Y, then the collection

D = {B C: B ε B and C ε C}

is a base for the topology of X Y.

Example 6. Consider the interpretation of this theorem for the case when X and Y are R, the set of real numbers. The open intervals on the real line constitute a base for the collection of all open sets of real numbers. Let U be an open interval (a, b) in X and V be an open interval (c, d) in Y. Then the collection of all open sets of the form U V is a base for the product topology on R R. See Fig. 5. Every open set of R R is the union of some of the members of this base. These open rectangles form a base for the product topology on R2, which is the usual topology on R2.

We thus see that while the definition gives as a base for a topology on R2 the collection of all products of open sets of R, the theorem provides us with a much smaller collection of all products (a, b) (c, d) of open intervals in R.

Def. Product space. Let {(Xi, Ti)} be a collection of topological spaces and let X

be the product of the sets Xi. The product set X with the product topology T is called the product topological space or simply the product space.

Theorem 3. Let X1, X2, .... , Xm be a finite number of topological spaces and let

X = X1 X2 .... Xm

be the product space. Then the collection of subsets

G1 G2 .... Gm ,

where Gi is an open subset of Xi, form a base for the product topology on X.

Subbase for product topology

Let X and Y be topological spaces. Let (x, y) be a point in the space X Y. The projection function π1: X Y X is, by definition,

π1(x, y) = x

and π2: X Y Y is

π2(x, y) = y .

The maps of π1 and π2 are called the projections of X Y onto X and Y, respectively.

If U is an open subset of X, then the set π1-1[U] is the set U Y, an open set of X Y (the set π1-1[U] is that subset of X Y that projects into U). See Fig. 6. Similarly, if V is an open subset of Y, then the set π2-1[V] is the set V X, also an open set of X Y (the set π2-1[V] is that subset of X Y that projects into V). See Fig. 7.

The intersection of these two sets U Y and V X is the set U V shown in Fig. 8.

Theorem 4. The collection

where U and V are open subsets in X and Y respectively is a subbase for the product topology on X Y.

Theorem 5. Let X1, X2, .... . , Xm be a set of topological spaces and let

X = X1 X2 .... Xm

be the product space. Then the collection of subsets

where Gi is an open subset of Xi is a subbase for the product topology on X. It is called the defining subbase for the product topology.

Since finite intersections of the subbase elements form a base for the topology we have:

Theorem 6. Let X1, X2, .... , Xm be a set of topological spaces and let

X = X1 X2 .... Xm

be the product space. Then the collection

where Gi is an open subset of Xi is a base for the product topology on X. It is called the defining base for the product topology.

General expression for πi-1[Gi] . The subset πi-1 [Gi] is that subspace of the product space that projects into the open set Gi . If we are considering two topological spaces X and Y the subspace of the product space X Y that projects into G1 is G1 Y where G1 is an open set in X. If we are considering three topological spaces X, Y and Z the subspace of the product space X Y Z that projects into G1 is G1 Y Z where G1 is an open set in X. See Fig. 9. The general formula for πi-1[Gi] for the case of m topological spaces X1, X2, .... , Xm is

Infinite sequences. Consider the case of an infinite but denumerable set of topological spaces X1, X2, X3, .... The product space

X = X1 X2 X3

then consists of all sequences

p = {a1, a2, a3, ...... }              where an ε Xn

In addition, if Gi is an open subset in Xi, then

Theorem 7. Let {(Xi, Ti)} be a collection of topological spaces and let X be the product of the sets Xi, i.e.

The coarsest topology T on X with respect to which all the projections πi: X Xi are continuous is the (Tychonoff) product topology.

We note that with the product topology, as it has been defined, all the projections are continuous since a function f is continuous if and only if the inverse of each open set in the range R of f is open in the domain D.

Theorem 8. For a Cartesian product of a finite number of topological spaces X1, X2, .... , Xn, a set is open in the product if and only if it is a product of sets U1, U2, ...... , Un , where Uk is open in Xk for each k. With this topology for the Cartesian product, it can be shown that the Cartesian product is compact if and only if each Xi is compact.

James & James. Mathematics Dictionary

Theorem 9. A function from a topological space Y into a product space

is continuous if and only if, for every projection πi: X Xi, the composition mapping

is continuous. See Fig. 10.

Theorem 10. Every projection πi: X Xi on a product space

is both open and continuous i.e. it is a bicontinuous mapping.

Theorem 11. A sequence p1, p2, p3, ..... of points in a product space

converges to the point q in X if and only if, for every projection πi: X Xi, the sequence πi(p1), πi(p2), πi(p3), ..... converges to πi(q) in the coordinate space Xi.

Functions viewed as infinite dimensional vectors. A function can be viewed as an infinite dimensional vector in the space . Let us consider the basis for this viewpoint. Let a function y = f(x) be defined on the interval [0, 1]. Divide the interval [0, 1] into equal sub-intervals with the points x1 = 0, x2, x3, .... , xn = 1 as shown in Fig. 11 and let

yi = f(xi) ,        i = 1, n

The function can thus be represented as the sequence of n numbers {y1, y2, ..... , yn }, an n-tuple, a point in n-dimensional space. Now let n and the function is represented as an infinite sequence {y1, y2, y3, .....}, a point in .

Let R1, R2, R3, .... be copies of R with the usual topology. Then the product space

consists of all sequences

p = {a1, a2, a3, ...... }              where ai ε Ri

If we equate infinite sequences with functions we see that the product space consists of the set of all real-valued functions.

We can state these ideas somewhat differently. Let Ri denote a copy of R. Conceive of the set {Ri} as being indexed by points in the closed interval A = [0, 1]. Then the product space is

A point p of the product space consists of a function y = f(x) i.e. an infinite sequence

p = {y1, y2, y3, .....}

In Fig. 12 is depicted a point p in . Each vertical line at a point i in the interval [0, 1] represents the coordinate space Ri . The value yi of the function at the point i is the i-th coordinate of p and corresponds to the projection of the point p on the coordinate set Ri .

Let us now describe one of the members of the defining subbase S for the product topology on . The subbase S consists of all of the subsets of of the form πi-1 [Gi] where Gi is an open subset of the coordinate set Ri . Suppose Gi is the open interval (2, 3). Then πi-1 [Gi] consists of all points p in such that ai ε Gi = (2, 3). Graphically, πi-1 [Gi] consists of all those functions passing through the open interval Gi = (2, 3) on the vertical line representing the coordinate set Ri. See Fig. 13.

Now let us describe one of the open sets of the defining base B for the product topology on .

Denote the open set by B. Then B is the intersection of a finite number of the members of the defining subbase S for the product topology, say,

B thus consists of all points p common to the three intervals of coordinates sets . Graphically, B consists of all functions passing through the open sets which lie on the vertical lines denoting the coordinate sets . See Fig. 14. One can visualize it as a bundle of fibers.

Def. Product invariant property.

A property P of a topological space is said to be product invariant if a product space

possesses P whenever each coordinate set Xi possesses P.

Tychonoff theorem. The product of compact spaces is compact.

Metric product spaces

Theorem 12. Let (X1, d1), (X2, d2), .......... ,(Xm, dm ) be metric spaces and let p = (a1, .... , am) and q = (b1, .... , bm) be arbitrary points in the product set

Then each of the following functions is a metric on the product set X:

Moreover, the topology on X induced by each of the above metrics is the product topology.

Theorem 13. Let (X1, d1), (X2, d2), .......... ,(Xm, dm ) be a denumerable collection of metric spaces and let p = (a1, a2, .... ) and q = (b1, b2 .... ) be arbitrary points in the product set

Then the function d defined by

is a metric on the product set X and the topology induced by d is the product topology.

References

1. Lipschutz. General Topology

2. Simmons. Introduction to Topology and Modern Analysis

3. Munkres. Topology, A First Course

4. James & James. Mathematics Dictionary