Iterating the E-field and H-field updates results in a marching-in-time process wherein sampled-data analogs of the continuous electromagnetic waves under consideration propagate in a numerical grid stored in the computer memory.

Iterating the same argument, r < sub > N − 1 </ sub > divides all the preceding remainders, including a and b. None of the preceding remainders r < sub > N − 2 </ sub >, r < sub > N − 3 </ sub >, etc.

Iterating the process produces a sequence of nested pentagons and pentagrams.

** Any union of countably many countable sets is itself countable.

That is, the union of countably many nowhere dense subsets of the space has empty interior.

Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space.

# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.

A σ-locally finite base is a base which is a union of countably many locally finite collections of open sets.

Consider the closed intervals for all integers k ; there are countably many such intervals, each has measure 1, and their union is the entire real line.

Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.

A construction adding at most countably many points is given in.

As a consequence of the Weierstrass approximation theorem, one can show that the space C is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients ; there are only countably many polynomials with rational coefficients.

Even a Turing oracle with random data is not computable ( with probability 1 ), since there are only countably many computations but uncountably many oracles.

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest.

It is possible to accommodate countably infinitely many coach-loads of countably infinite passengers each.

The probability measure function must satisfy a simple requirement: the probability of a union of two ( or countably many ) disjoint events must be equal to the sum of probabilities of each of these events.

* The ( pointwise ) supremum, infimum, limit superior, and limit inferior of a sequence ( viz., countably many ) of real-valued measurable functions are all measurable as well.

Sets that can be constructed as the union of countably many closed sets are denoted F < sub > σ </ sub > sets.

For our purposes, this means that yes / no questions that depend on countably many coordinates have a probabilistic answer.

* f can only have countably many discontinuities in its domain.

Every tree with only countably many vertices is a planar graph.

* Every connected graph with only countably many vertices admits a normal spanning tree.

The union of countably many nowhere dense sets, however, need not be nowhere dense.

The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.

In the bivariate case, we also have a theorem that makes the first equivalent condition for multivariate normality less restrictive: it is sufficient to verify that countably many distinct linear combinations of X and Y are normal in order to conclude that the vector is bivariate normal.

Since the poles of a meromorphic function are isolated, there are at most countably many.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.

* For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order p < sup > n </ sup > for each positive integer n ( and is in fact the union of these copies ).

For example, in the topological space ω < sub > 1 </ sub >+ 1, the element ω < sub > 1 </ sub > is in the closure of the subset ω < sub > 1 </ sub > even though no sequence of elements in ω < sub > 1 </ sub > has the element ω < sub > 1 </ sub > as its limit: an element in ω < sub > 1 </ sub > is a countable set ; for any sequence of such sets, the union of these sets is the union of countably many countable sets, so still countable ; this union is an upper bound of the elements of the sequence, and therefore of the limit of the sequence, if it has one.

A multiplication operator is defined as follows: Let be a countably additive measure space and f a real-valued measurable function on X.

It shows that a compact operator K on an infinite-dimensional Banach space has spectrum that is either a finite subset of C which includes 0, or the spectrum is a countably infinite subset of C which has 0 as its only limit point.

The language of propositional modal logic consists of a countably infinite set of propositional variables, a set of truth-functional connectives ( in this article and ), and the modal operator (" necessarily ").

It must assign 0 to the empty set and be ( countably ) additive: the measure of a " large " subset that can be decomposed into a finite ( or countable ) number of " smaller " disjoint subsets, is the sum of the measures of the " smaller " subsets.

Any topological space which is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself.

For example, every subset of Cantor or Baire space is a set ( that is, a set which equals the intersection of countably many open sets ).

It's sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a Hilbert space with countably infinite dimension.

The meagre subsets of a fixed space form a sigma-ideal of subsets ; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.

Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.

That theorem implies that there is some countably infinite subset of the real numbers, whose members we will call internal numbers, and some countably infinite collection of sets of internal numbers, whose members we will call " internal sets ", such that the domain consisting of internal numbers and internal sets satisfies exactly the same first-order sentences satisfied as the domain of real numbers and sets of real numbers.

More precisely, a right Haar measure on a locally compact group G is a countably additive measure μ defined on the Borel sets of G which is right invariant in the sense that μ ( A x ) = μ ( A ) for x an element of G and A a Borel subset of G and also satisfies some regularity conditions ( spelled out in detail in the article on Haar measure ).

Any discrete subset of Euclidean space is countable, since the isolation of each of its points ( together with the fact the rationals are dense in the reals ) means that it may be mapped 1-1 to a set of points with rational co-ordinates, of which there are only countably many.

It also suffices to prove that every infinite set is Dedekind-infinite ( equivalently: has a countably infinite subset ).

If Σ is a sigma-algebra, then the space is defined as the subset of consisting of countably additive measures.

* There exists a countable subset ( finite or countably infinite ) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form