Continuum - straight lines, real numbers

Real Line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one. It can be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.

Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, {\displaystyle \mathbb {R} } \mathbb {R} , the letter “R” in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space.

This article focuses on the aspects of R as a geometric space in topology, geometry, and real analysis. The real numbers also play an important role in algebra as a field, but in this context R is rarely referred to as a line. For more information on R in all of its guises, see real number.

Number line

In basic mathematics, a number line is a picture of a graduated straight line that serves as abstraction for real numbers, denoted by {\displaystyle \mathbb {R} } \mathbb {R} . Every point of a number line is assumed to correspond to a real number, and every real number to a point.[1]

The integers are often shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from -9 to 9, the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.

Linear continuum

In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.

Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound. More symbolically:

S has the least-upper-bound property, and For each x in S and each y in S with x < y, there exists z in S such that x < z < y

A set has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound. Linear continua are particularly important in the field of topology where they can be used to verify whether an ordered set given the order topology is connected or not.[1]

Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum.