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Page "Compact space" ¶ 67
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For and any
For the answer cannot be derived from any socially cohesive element in the disrupting community.
For the truth formerly experienced by the community no longer has existential status in the community, nor does any answer elaborated by philosophers or theoriticians.
William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.
For if Serenissimus made the sign of the Cross with his right hand, and meant it, with his left he beckoned lewdly to any lady who happened to catch his eye.
For them only a little more needed to be learned, and then all physical knowledge could be neatly sorted, packaged and put in the inventory to be drawn on for the solution of any human problem.
For one thing, there wasn't going to be any ceremony at all this year.
For that reason any democratic reform and effort to bring genuine representative government to the Dominican Republic will need the greatest sympathy and help.
For those communities which have financial difficulties in effecting adjustments, there are a number of alternatives any one of which alone, or in combination with others, would minimize if not even eliminate the problem.
For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of $200 million.
For the making of selections on the basis of excellence requires that any foundation making the selections shall have available the judgments of a corps of advisors whose judgments are known to be good: such judgments can be known to be good only by the records of those selected, by records made subsequent to their selection over considerable periods of time.
For the near term, however, it must be realized that the industrial and commercial market is somewhat more sensitive to general business conditions than is the military market, and for this reason I would expect that any gain in 1961 may be somewhat smaller than those of recent years ; ;
For any house.
For proper accreditation of schools, teachers in any course must have a degree at least one level above that for which the student is a candidate.
For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.
For the lines of any plane, **yp, meeting Q in a conic C, are transformed into the congruence of secants of the curve C' into which C is transformed in the point involution on Q.
For any pencil in a plane containing a Af-fold secant of **zg has an image regulus which meets the plane of the pencil in Af lines, namely the images of the lines of the pencil which pass through the intersection of **zg and the multiple secant, plus an additional component to account for the intersections of the images of the general lines of the pencil.
For any choice of admissible policy Af in the first stage, the state of the stream leaving this stage is given by Af.
For in the modern world neither `` spirit '' nor `` matter '' refer to any generally agreed-upon elements of experience.
For in Christ Jesus neither circumcision nor uncircumcision but a new creation is of any account.
For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.
: For any set X of nonempty sets, there exists a choice function f defined on X.
: For any set A, the power set of A ( with the empty set removed ) has a choice function.
: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

For and subset
For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.
For example, the term is used to describe systems such as verlan and louchébem, which retain French syntax and apply transformations only to individual words ( and often only to a certain subset of words, such as nouns, or semantic content words ).
For example, in the mid-Atlantic US, the X-Acto name is likelier to evoke only a specific subset of these knives ( the pencil-shaped hobby knife ), which may explain why the " utility knife " name, with its specificity, is more common there for the larger type.
For instance, a diagonal operator on the Hilbert space may have any compact nonempty subset of C as spectrum.
For instance, does a subset of the set
For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.
For the converse direction, let A be a partially ordered set and T a totally ordered subset of A.
For any subset N of M, the monoid N * is the smallest monoid that contains N.
For example, a good displays local network effects when rather than being influenced by an increase in the size of a product's user base in general, each consumer is influenced directly by the decisions of only a typically small subset of other consumers, for instance those he or she is " connected " to via an underlying social or business network.
For these and other reasons, most sources consider C-H compounds to be only a subset of " organic " compounds.
For determinantal rank, the statement is that if the row rank ( column rank ) of a matrix is p, then one can choose a p × p submatrix that is invertible: a subset of the rows and a subset of the columns simultaneously define an invertible submatrix.
For example, an unbiased sample of Australian men taller than 2m might consist of a randomly sampled subset of 1 % of Australian males taller than 2m.
For this closure is characterized in terms of limits of filter bases: if Y is a subset of X and z is a point of X, then z is in the closure of Y if and only if there exists a filter base B consisting of subsets of Y which converges to z.
For example, in the United Kingdom, the executive forms a subset of the legislature, as did — to a lesser extent — the judiciary until the establishment of the Supreme Court of the United Kingdom.
For any topological space X, let C ( X ) denote the family of real-valued continuous functions on X and let C *( X ) be the subset of bounded real-valued continuous functions.
For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.
For example, the set of rational numbers — those numbers which can be written as a quotient of integers — contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.
For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape formed by a rubber band stretched around X.
For example, the product of the unit circle ( with its usual topology ) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset
For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S ( this point may be x itself ).
For instance, the general linear group GL ( n, R ) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL ( n, R ) as a subset of Euclidean space R < sup > n × n </ sup >.
For a collection T of subsets of X ( that is, for any subset of the power set P ( X ) of X ), let

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