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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
C^{1} the interior and boundary of the unit circle. Then
I
I is the unit square, C^{1}
I is a cylinder and
C^{1}
C^{1} is a torus. See Fig. 3.

The Cartesian product
A_{1}
A_{2}
A_{n} . The Cartesian
product A_{1}
A_{2}
A_{n} of n sets A_{1},
A_{2}, ...... , A_{n} is the set of all ordered n-tuples (a_{1}, a_{2}, .... , 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 (a_{1}, a_{2},
.... , a_{n}) as a point in an n-dimensional
space defined by the Cartesian product
A_{1}
A_{2}
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 (a_{1}, a_{2}, .... , a_{n}) is called the i-th coordinate of the vector (a_{1}, a_{2}, .... ,
a_{n}) in this n-dimensional space. This i-th coordinate of the point is called the projection of the
vector (a_{1}, a_{2}, .... , a_{n}) onto the i-th coordinate set A_{i}.

The above represents a generalization of concepts associated with the usual 3-space R^{3} and n-space R^{n}. The space R^{n} becomes a special case of the above when A_{1 }= A_{2} = .... = A_{n} = R.

Notations. The Cartesian product A_{1}
A_{2}
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 = (a_{1}, a_{2}, .... , a_{n}) be a point in the n-dimensional space defined by the Cartesian product
A_{1}
A_{2}
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 = (a_{1}, a_{2}, .... , a_{n}). Here a_{i} represents the projection
of the vector (a_{1}, a_{2}, .... , 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
A_{1}
A_{2}
A_{n } into the i-th coordinate set X_{i}.

Example 4. Let R_{1}, R_{2} and R_{3} denote copies of R. Consider the point X = (3.1, 6.5, 2.8) in
three dimensional space R^{3} = R
R
R = R_{1}
R_{2}
R_{3}. Then

π_{1}(X) = 3.1 is the projection of X in R_{1}.

π_{2}(X) = 6.5 is the projection of X in R_{2}.

π_{3}(X) = 2.8 is the projection of X in R_{3}.

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 R^{2}, R^{3}, .... , R^{n}. The Cartesian product R
R is usually denoted by R^{2}, the
Cartesian product R
R
R is usually denoted by R^{3} and the Cartesian product
R
R
.....
R (n times) consisting of all n-tuples (x_{1}, x_{2}, .... , x_{n}) is usually denoted by R^{n}.

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 A_{1}
A_{2}
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 C_{1} 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 R^{2}, which is the
usual topology on R^{2}.

We thus see that while the definition gives as a base for a topology on R^{2} 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 X_{1}, X_{2}, .... , X_{m} be a finite number of topological spaces and let

X = X_{1 }
X_{2 }
....
X_{m}

be the product space. Then the collection of subsets

G_{1}
G_{2}
....
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
π_{1}: X
Y
X is, by definition,

π_{1}(x, y) = x

and π_{2}: X
Y
Y is

π_{2}(x, y) = y .

The maps of π_{1} and π_{2 }are called the projections of
X
Y onto X and Y, respectively.

If U is an open subset of X, then the set π_{1}^{-1}[U] is
the set U
Y, an open set of X
Y (the set π_{1}^{-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 π_{2}^{-1}[V] is the set V
X, also an
open set of X
Y (the set π_{2}^{-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 X_{1}, X_{2}, ...._{ }. , X_{m} be a set of
topological spaces and let

X = X_{1 }
X_{2 }
....
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 X_{1}, X_{2}, .... , X_{m} be a
set of topological spaces and let

X = X_{1 }
X_{2 }
....
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 G_{1} is G_{1}
Y where G_{1} 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 G_{1}
is G_{1}
Y
Z where G_{1} 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 X_{1}, X_{2}, .... , X_{m} is

Infinite sequences. Consider the case of an infinite but denumerable set of topological
spaces X_{1}, X_{2}, X_{3}, ...._{ } The product space

X = X_{1 }
X_{2 }
X_{3}

then consists of all sequences

p = {a_{1}, a_{2}, a_{3}, ...... } 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 X_{1}, X_{2}, .... , X_{n, }a
set is open in the product if and only if it is a product of sets U_{1}, 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 p_{1}, p_{2}, p_{3}, ..... 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}(p_{1}), π_{i}(p_{2}), π_{i}(p_{3}), ..... 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 x_{1} = 0, x_{2}, x_{3}, .... , 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 {y_{1}, y_{2}, ..... , y_{n} }, an n-tuple, a
point in n-dimensional space. Now let n
and the function is represented as an infinite
sequence {y_{1}, y_{2}, y_{3}, .....}, a point in
.

Let R_{1}, R_{2}, R_{3}, .... be copies of R with the usual topology. Then the product space

consists of all sequences

p = {a_{1}, a_{2}, a_{3}, ...... } 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 = {y_{1}, y_{2}, y_{3}, .....}

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 (X_{1}, d_{1}), (X_{2}, d_{2}), .......... ,(X_{m}, d_{m }) be metric spaces and let p = (a_{1}, .... , a_{m})
and q = (b_{1}, .... , 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 (X_{1}, d_{1}), (X_{2}, d_{2}), .......... ,(X_{m}, d_{m }) be a denumerable collection of metric
spaces and let p = (a_{1}, a_{2}, .... ) and q = (b_{1}, b_{2 }...._{ }) 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

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