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

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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 A_n . The Cartesian product A₁ A₂ A_n of n sets A₁, A₂, ...... , A_n is the set of all ordered n-tuples (a₁, a₂, .... , a_n) for a_i ε A_i, where each a_i assumes all the values in A_i, i = 1, 2, ...., n. We can view an n-tuple (a₁, a₂, .... , a_n) as a point in an n-dimensional space defined by the Cartesian product A₁ A₂ A_n . The component sets A_1, A_2, , A_n of the product are called the coordinate sets of the space. The set A_i is called the i-th coordinate set of the product. The i-th component of the n-tuple (a₁, a₂, .... , a_n) is called the i-th coordinate of the vector (a₁, a₂, .... , a_n) in this n-dimensional space. This i-th coordinate of the point is called the projection of the vector (a₁, a₂, .... , a_n) onto the i-th coordinate set A_i.

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 = .... = A_n = R.

Notations. The Cartesian product A₁ A₂ A_i of an indexed collection of sets {A_i}_{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, .... , a_n) be a point in the n-dimensional space defined by the Cartesian product A₁ A₂ A_n. The projection function π_i(X) is defined as

π_i(X) = a_i

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

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, .... , x_n) 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 A_n 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 {(X_i, T_i)} be a collection of topological spaces and let X

be the product of the sets X_i. 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, .... , X_m be a finite number of topological spaces and let

X = X₁ X₂ .... X_m

be the product space. Then the collection of subsets

G₁ G₂ .... G_m ,

where G_i is an open subset of X_i, 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 π₂are 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, ..... , X_m be a set of topological spaces and let

X = X₁ X₂ .... X_m

be the product space. Then the collection of subsets

where G_i is an open subset of X_i 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, .... , X_m be a set of topological spaces and let

X = X₁ X₂ .... X_m

be the product space. Then the collection

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

General expression for π_i^-1[G_i] . The subset π_i^-1[G_i] is that subspace of the product space that projects into the open set G_i. 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 π_i^-1[G_i] for the case of m topological spaces X₁, X₂, .... , X_m 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 a_n ε X_n

In addition, if G_i is an open subset in X_i, then

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

The coarsest topology T on X with respect to which all the projections π_i: X X_i 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, .... , X_n,a set is open in the product if and only if it is a product of sets U₁, U_{2, ...... ,} U_n , where U_k is open in X_k 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 X_i 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 π_i: X X_i, the composition mapping

is continuous. See Fig. 10.

Theorem 10. Every projection π_i: X X_i 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 π_i: X X_i, the sequence π_i(p₁), π_i(p₂), π_i(p₃), ..... converges to π_i(q) in the coordinate space X_i.

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, .... , x_n = 1 as shown in Fig. 11 and let

y_i = f(x_i) , i = 1, n

The function can thus be represented as the sequence of n numbers {y₁, y₂, ..... , y_n }, 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 a_i ε R_i

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 R_i denote a copy of R. Conceive of the set {R_i} 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 R_i . The value y_i 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 R_i .

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 π_i^-1[G_i] where G_i is an open subset of the coordinate set R_i . Suppose G_i is the open interval (2, 3). Then π_i^-1[G_i] consists of all points p in such that a_i ε G_i = (2, 3). Graphically, π_i^-1[G_i] consists of all those functions passing through the open interval G_i = (2, 3) on the vertical line representing the coordinate set R_i. 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 X_i possesses P.

Tychonoff theorem. The product of compact spaces is compact.

Metric product spaces

Theorem 12. Let (X₁, d₁), (X₂, d₂), .......... ,(X_m, d_m) be metric spaces and let p = (a₁, .... , a_m) and q = (b₁, .... , b_m) 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), .......... ,(X_m, d_m) 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