Indeed, naive set theory might be said to be based on this notion.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

Another way to describe G-sets is the functor category, where is the groupoid ( category ) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X = F and for every g in G ( i. e. for every morphism in ) induces a bijection F < sub > g </ sub >: X → X.

Indeed, the Royals set a franchise low with a. 398 winning percentage ( 64 – 97 record ) in 1999, and lost 97 games again in 2001.

Indeed, Jesse Lasky Jr., the co-writer on The Ten Commandments, described how the director would customarily spread out prints of Alma-Tadema paintings to indicate to his set designers the look he wanted to achieve.

* " Indeed men too often take upon themselves in the prosecution of their revenge to set the example of doing away with those general laws to which all can look for salvation in adversity, instead of allowing them to subsist against the day of danger when their aid may be required " ( Strassler 201 / 3. 84. 3 ).

Indeed, a totally ordered set ( with its order topology ) is compact as a topological space if it is complete as a lattice.

Indeed, a set is nowhere dense if and only if its closure is nowhere dense.

Indeed, some novels go so far as to posit a love-affair between Matilda and Stephen, e. g. the Janna Mysteries by Felicity Pulman, set during the civil war between Stephen and Matilda.

Indeed, the Mandelbrot set is defined as the set of all such that is connected.

Observe first the following: if a is a limit point of S, then any finite collection C of open sets, such that each open set U ∈ C is disjoint from some neighborhood V < sub > U </ sub > of a, fails to be a cover of S. Indeed, the intersection of the finite family of sets V < sub > U </ sub > is a neighborhood W of a in R < sup > n </ sup >.

Indeed, he is, in many ways, the father of Dominican democracy " and Jimmy Carter complimented him saying " President Balaguer has set an example for all leaders in this nation in changing his own country and his own people away from a former totalitarian government to one of increasingly pure democracy.

Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem ( BPIT ), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory ( ZF ) and the ZF theory augmented by the axiom of choice ( ZFC ).

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.

Indeed cases of this have been documented so frequently over the last decade that a term has been given for this set of complication: neurologic serious adverse events ( SAEs ).

Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by adding its limit points.

Indeed, it was Eileen, in league with the sinister Mike Shaw ( Tony O ' Callaghan ), who had set the fire in the internet café that killed Roy.

Indeed the 2008 closure of “ Tempelhof International Airport ” and 2012 closure of “ Berlin-Tegel International Airport ” were set as a prerequisite for opening BER to traffic.

Indeed, its operation was a lesson in engineering simplicity, easy to set up, easy to calibrate, and easy to use, with only one rewind and forward operation in an utterly unique unidirectional auto-reverse deck.

Indeed, the entire breast has a network of nerve endings, and it has the same number of nerve endings no matter how large it is.

Indeed, Chao characterized GC as having " the initial consonants of the Wu dialects [...], the vowels of Mandarin, and the endings of Cantonese.

Indeed, a lighter weight works much better because a greater, more extensive split can be performed.

Indeed, one can assume that exactly one parallel through a point outside a line exists, or that infinitely many exist.

Indeed, the International Committee for Weights and Measures ( CIPM ) notes that " its definition applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored.

Indeed, Disraeli had objected to Murray about Croker inserting " high Tory " sentiment, writing that " it is quite impossible that anything adverse to the general measure of Reform can issue from my pen.

Indeed, one such clause can be rewritten as, that is, if are all true, then y needs to be true as well.

Indeed it can be argued the country has never been independent since there is an arguable legitimate succession of states, systems and entities from the Norman Conquest, 1066.

Indeed, it turns out that numerous important constructions can be described in a purely categorical way.

( Indeed, every production of a context free grammar is of the form V → w where V is a single nonterminal symbol, and w is a string of terminals and / or nonterminals ( w can be empty )).

Indeed, the symbiosis with root fungi is quite common among the order representatives, and there are even three kinds of it which can be found exclusively among Ericales ( namely, ericoid, arbutoid and monotropoid mycorrhiza ).

Indeed, Winograd showed that the DFT can be computed with only irrational multiplications, leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes ; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.

Indeed it can be shown that any Q-automorphism must preserve the ordering of the real numbers and hence must be the identity.

Indeed, the importance of the concepts of good and evil can be regarded as distinguishing mark between high fantasy and sword and sorcery.

Indeed, an intuitive summary of inertial frames can be given as: In an inertial reference frame, the laws of mechanics take their simplest form.

Indeed, the entrenched nature of command-and-control can be seen in the way that recent layoffs have been conducted with management ranks affected far less than employees at the lower levels.

Indeed, this can be extended to the observation that introducing a DNA-binding sequence within the nucleosome increases the accessibility of adjacent regions of DNA when bound.

Indeed, he argues that any course of action can be made out to accord with some particular rule, and that therefore a rule cannot be used to explain an action.

Indeed, the renowned physicist Richard Feynman once said, " I think I can safely say that nobody understands quantum mechanics.

Indeed Ingram's influence on Powell's later work can be detected, especially in its themes in illusion, dreaming, magic and the surreal.

Indeed, the scientific method, and the scientists themselves, can be prepared in many other ways to harness luck and make discoveries.

Indeed Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as Huygens ' principle.

i. e., z is a vector orthogonal to the vector v ( Indeed, z is the projection of u onto the plane orthogonal to v .) We can thus apply the Pythagorean theorem to