Concepts

This is a working list of mathematical concepts that are used several times on this blog, and require a concise description for convenience. They are listed alphabetically below. Scroll down to find a topic.

Continuity
A function is called continuous, if, intuitively, it has no "jumps". In other words, as one approaches a point in the domain of the function from any direction, the outputs must also approach the value of the function at the point. If this property is not satisfied, the function in question is discontinuous. More formally, if f is a function defined on the set S into the set T, on which there are distance functions D1 and D2 on S and T, respectively, which assign a nonnegative value to each pair of points in the set on which they are defined, then f is continuous at a point xS if and only if, for any ε > 0, there exists a δ > 0 such that for D1(x,y) < δ and yS imply that D2(f(x),f(y)) < ε.
Examples:
The function y = x on the real numbers is continuous.
The function f defined on the real numbers as f(x) ≡ -1 for negative x and f(x) ≡ 0 for nonnegative x, is discontinuous at x = 0. As once approaches 0 from the left, the function will suddenly "jump" from -1 to 1 without taking any values in between.

Convergence
Convergence, when describing a sequence of numbers, refers to the property that the members of the sequence become closer and closer to a fixed value. In formal terms, a sequence, denoted {an}, which contains members a1, a2,..., converges to a value a, in a space on which there is a distance function D, if, for every ε > 0 there exists a positive integer N such that n > N implies D(an,a) < ε. In other words, a sequence converges if, for every possible radius ε, the "circle" of this radius centered at a contains the "rest" of the sequence after some term aN. If a sequence does not converge, it is said to diverge, or be divergent.
Examples:
The sequence {1,2,3,4,...} diverges because the terms are not bounded; they go to infinity rather than converging.
The sequence {0,1,0,1,...} is bounded, but still diverges. This is because, in the definition above, if we choose ε to equal 1/2, there is no point in the sequence in which all of the rest of the terms are contained within an interval of length 1/2, as there are terms differing by 1 arbitrarily far into the series.
The sequence {1/2, 3/4, 7/8, 15/16,...} converges to the value 1. For any positive number ε in the above definition, there is an N such that 1/(2N) < ε. Since all an such that n > N satisfy |an - 1| < 1/(2N) < ε, the conditions of the definition are fulfilled and the sequence converges to 1.

Countable Set
A countable set is a type of infinite set. A set is countable if its members can be put into a one-to-one correspondence with the natural numbers. More simply, a set of countable if its elements can be labeled 1, 2, 3, and so on through all natural numbers without "running out" of labels. If an infinite set is not countable, it is called uncountable.
Examples:
The whole numbers ({0,1,2,...}) are countable, with the labeling system 0→1, 1→2,... (i.e., 0 is labeled 1, 1 is labeled 2, and every member has a corresponding number)
The integers ({...-1,0,1,...}) are countable
The rational numbers are countable
The real numbers are uncountable
The complex numbers are uncountable
All of the sets above are proved to be countable or uncountable in their respective cases in the Infinity Series.

Dense Set
A dense set is type of set. A set X is dense in the set Y if X is a subset of Y, and, for any member of Y, say y, either y itself is a member of X or there is a member of X arbitrarily close to y. If the ambient set Y has a notion of distance between two of its members x and y, i.e. a function D that assigns a nonnegative real number to every pair of members of Y, then the concept of denseness can be stated more formally:
A subset X of a set Y is dense in Y if, given a distance function D defined on all pairs of members of Y, every member of Y is either
1. a member of X, or
2. there exists, for every positive real number ε, no matter how small, a member x of X such that D(x,y) < ε.
In simpler terms, the above means that any member of Y can be approximated arbitrarily closely by a member of X. For instance, the rational numbers are dense in the real numbers because any real number, for example π, can be approximated to any degree of accuracy by its decimal expansion: 3.141592...

Field
A field is a set, together with an addition operation + and a multiplication operation * defined for every pair of its members such that the set is closed under these operations, addition is commutative and associative, multiplication is associative, multiplication can be distributed over addition, every member of the set has a unique additive and multiplicative inverse, there is a additive identity 0, and there is a multiplicative identity 1.
To illustrate this concept, consider the following example: The rational numbers, Q, form a field, with addition and multiplication defined in the normal manner. That the set Q is closed under these operations means that, for any two rational numbers a and b, a + b and a*b are both rational numbers. Certainty addition is commutative and associative and multiplication is associative among rational numbers (in fact, multiplication is also commutative, that is a*b = b*a, but this is not required for a field). Multiplication also distributes over addition (a*(b + c) = a*b + a*c for rational a, b, and c). The additive inverse of a rational number a is -a, and 0 + a = a, making 0 the additive identity. Similarly, the multiplicative inverse of a is 1/a, clearly also a rational number, and a*1 = a, making 1 the multiplicative identity. Thus Q is a field.

Manifold
A manifold, simply speaking, is a surface of some dimension that satisfies certain properties. A manifold of dimension m is a surface, which, at small scales, "resembles" the Euclidean, or flat space Rm (i.e., R1 is a line, R2 is a plane, and R3 is what one usually thinks of by "space").
To see how this "resemblance" comes about, consider the circle (by "circle" is meant the curve only, not its interior). By zooming in sufficiently, a portion of a circle looks like a line. Therefore, the circle is a manifold of dimension 1.