7.2 Sets and Spaces
389
relations, which is called the giving the set some space structure, and the set given
different space structures is called a different space. The space composed of the sets
of numbers is called the number space. The space composed of the sets of functions
is called the function space. The set introduced some kind of relationship between
elements is called the abstract space. If in a set two kinds of algebraic operations of
addition and multiplication can be done, then the set is called the linear space structure. The set given the linear space structure is called the linear space. If defining
the concept of distance between two elements in a set, then which is called giving
the set a distance structure.
Let X be a nonempty set, ρ is a real-valued function defined on X × X , if
for arbitrary two elements x and y in X , there corresponds a real number ρ(x, y)
according to certain rules, and ρ(x, y) satisfies the following three distance axioms
or metric axioms: (1) The positive definiteness and identity: ρ(x, y) 0, and the
necessary and sufficient condition of ρ(x, y) = 0 is x = y; (2) The symmetry:
ρ(x, y) = ρ(y, x); (3) The triangle inequality: ρ(x, y) + ρ(x, z) ρ(y, z) is valid
for arbitrary three elements x, y and z in X , then ρ(x, y) is called the metric or
distance between x and y on X , and X is called the metric space with distance
ρ(x, y), or it is called the distance space, it is denoted by (X, ρ). Sometimes X is
also called the basic set or fundamental set of metric space. In the case of the metric
ρ not confused, the metric space (X, ρ) can is denoted by X for short. The elements
x, y, … in the metric space X are called the point in X . Thus, the so-called metric
space is that the distance is introduced in the set X . In a set, the definition way of
the distance is not unique. For the same set X , if the introduced distance is different,
then the constituted metric space is also different. After the distance in the set X
is introduced, which is called that a topological structure is introduced in X . The
set with topological structure is called a topological space. Using the topological
structure, which can compare the distance between the elements in a set, perform
limit operation.
In the above mentioned definition, the geometric significance of triangle inequality
is obvious, it represents the fact that the sum of two sides is greater than a third side
in a triangle.
For the nonempty subset S of the metric space (X, ρ), if still taking the distance
ρ in X as the distance in S, then S is also a metric space, and S is called the metric
subspace of X , it is called the subspace for short. Clearly, (S, ρ) ⊂ (X, ρ).
Let A, B be two sets in the metric space (X, ρ), if for any x ∈ A, there is a number
sequence {y n } ∈ B, such that lim
n→∞
y n = x, then B is called the dense set in A, it is
called the dense in A for short. If A = X , then B is called the dense set in X , it is
called dense in X for short.
Let {x n } (n = 1, 2, · · · ) be a number sequence in the metric space (X, ρ), if there
is x ∈ X , such that lim
n→∞
ρ(x n , x) = 0, then {x n } is called the convergent sequence
of numbers or convergent sequence, x is called the limit of {x n }, it is denoted by
lim
n→∞
x n = x. Note that the convergence of the number sequence {x n } in the metric
space (X, ρ) means that the limit x of {x n } not only exists, but also x ∈ X .
389
relations, which is called the giving the set some space structure, and the set given
different space structures is called a different space. The space composed of the sets
of numbers is called the number space. The space composed of the sets of functions
is called the function space. The set introduced some kind of relationship between
elements is called the abstract space. If in a set two kinds of algebraic operations of
addition and multiplication can be done, then the set is called the linear space structure. The set given the linear space structure is called the linear space. If defining
the concept of distance between two elements in a set, then which is called giving
the set a distance structure.
Let X be a nonempty set, ρ is a real-valued function defined on X × X , if
for arbitrary two elements x and y in X , there corresponds a real number ρ(x, y)
according to certain rules, and ρ(x, y) satisfies the following three distance axioms
or metric axioms: (1) The positive definiteness and identity: ρ(x, y) 0, and the
necessary and sufficient condition of ρ(x, y) = 0 is x = y; (2) The symmetry:
ρ(x, y) = ρ(y, x); (3) The triangle inequality: ρ(x, y) + ρ(x, z) ρ(y, z) is valid
for arbitrary three elements x, y and z in X , then ρ(x, y) is called the metric or
distance between x and y on X , and X is called the metric space with distance
ρ(x, y), or it is called the distance space, it is denoted by (X, ρ). Sometimes X is
also called the basic set or fundamental set of metric space. In the case of the metric
ρ not confused, the metric space (X, ρ) can is denoted by X for short. The elements
x, y, … in the metric space X are called the point in X . Thus, the so-called metric
space is that the distance is introduced in the set X . In a set, the definition way of
the distance is not unique. For the same set X , if the introduced distance is different,
then the constituted metric space is also different. After the distance in the set X
is introduced, which is called that a topological structure is introduced in X . The
set with topological structure is called a topological space. Using the topological
structure, which can compare the distance between the elements in a set, perform
limit operation.
In the above mentioned definition, the geometric significance of triangle inequality
is obvious, it represents the fact that the sum of two sides is greater than a third side
in a triangle.
For the nonempty subset S of the metric space (X, ρ), if still taking the distance
ρ in X as the distance in S, then S is also a metric space, and S is called the metric
subspace of X , it is called the subspace for short. Clearly, (S, ρ) ⊂ (X, ρ).
Let A, B be two sets in the metric space (X, ρ), if for any x ∈ A, there is a number
sequence {y n } ∈ B, such that lim
n→∞
y n = x, then B is called the dense set in A, it is
called the dense in A for short. If A = X , then B is called the dense set in X , it is
called dense in X for short.
Let {x n } (n = 1, 2, · · · ) be a number sequence in the metric space (X, ρ), if there
is x ∈ X , such that lim
n→∞
ρ(x n , x) = 0, then {x n } is called the convergent sequence
of numbers or convergent sequence, x is called the limit of {x n }, it is denoted by
lim
n→∞
x n = x. Note that the convergence of the number sequence {x n } in the metric
space (X, ρ) means that the limit x of {x n } not only exists, but also x ∈ X .