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 complete, i.e., which "lacks gaps" in the sense that every nonempty 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 in the set. 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. Unlike the standard real line, a linear continuum may be bounded on either side: for example, any (real) closed interval is a linear continuum. The ordered set of real numbers, R, with its usual order is a linear continuum, and is the archetypal example. Property b) is trivial, and property a) is simply a reformulation of the completeness axiom. Examples in addition to the real numbers: sets which are order-isomorphic to the set of real numbers, for example a real open interval, and the same with half-open gaps (note that these are not gaps in the above-mentioned sense) the affinely extended real number system and order-isomorphic sets, for example the unit interval the set of real numbers with only +∞ or only −∞ added, and order-isomorphic sets, for example a half-open interval the long line The set I × I (where × denotes the Cartesian product and I = [0, 1]) in the lexicographic order is a linear continuum. Property b) is trivial. To check property a), we define a map, π1 : I × I → I by π1 (x, y) = x This map is known as the projection map. The projection map is continuous (with respect to the product topology on I × I) and is surjective.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (4)

Related courses (1)

Related lectures (1)

Completeness of the real numbers

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Long line (topology)

In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties (e.g., it is neither Lindelöf nor separable). Therefore, it serves as an important counterexample in topology. Intuitively, the usual real-number line consists of a countable number of line segments laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

MATH-318: Set theory

Set Theory as a foundational system for mathematics. ZF, ZFC and ZF with atoms. Relative consistency of the Axiom of Choice, the Continuum Hypothesis, the reals as a countable union of countable sets,

Covers the basics of sets and operations in mathematics, from set properties to advanced operations.