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.

One of the basic tenets of Euclidean geometry is that two figures ( that is, subsets ) of the plane should be considered equivalent ( congruent ) if one can be transformed into the other by some sequence of translations, rotations and reflections.

Many interrelationships of these areas exist, and some are subsets of others.

Violence has also sometimes appeared in the punk subculture, but has been opposed by some subsets of the subculture, such as the pacifist strain of anarcho-punk.

Urban explorers often attempt some or all of these subsets of urban exploration.

Not every subset of the sample space Ω must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be “ measured ”.

Compared to other dense subsets of the real line, such as the rational numbers, the dyadic rationals are in some sense a relatively " small " dense set, which is why they sometimes occur in proofs.

Because mitochondrial disorders contain many variations and subsets, some particular mitochondrial disorders are very rare.

Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of integers.

Most of these receptors are not unique to NK cells and can be present in some T cell subsets as well.

Yen's improvement first assigns some arbitrary linear order on all vertices and then partitions the set of all edges into one of two subsets.

Cluster analysis is the assignment of a set of observations into subsets ( called clusters ) so that observations within the same cluster are similar according to some predesignated criterion or criteria, while observations drawn from different clusters are dissimilar.

( If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.

A subset V of A < sup > n </ sup > is called an affine algebraic set if V = Z ( S ) for some S. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.

Suppose we have some such family of subsets A.

For 1 ≤ p ≤ ∞ these real Hardy spaces H < sup > p </ sup > are certain subsets of L < sup > p </ sup >, while for p < 1 the L < sup > p </ sup > spaces have some undesirable properties, and the Hardy spaces are much better behaved.

For example, in the natural subset-inclusion ordering on some powerset, any infinite element ( i. e. set ) is much more " informative " than any of its finite subsets.

Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model.

Adding up some subsets of its divisors ( e. g., 5, 10 and 15 ) gives 30, hence 30 is a semiperfect number.

Adding up some subsets of its divisors ( e. g., 1, 4, 5, 10 and 20 ) gives 40, hence 40 is a semiperfect number.

Adding up some subsets of its divisors ( e. g., 1, 4, 5, 10, 20 and 40 ) gives 80, hence 80 is a semiperfect number.

However, if f has domain S and g has domain T, both subsets of X, then f and g are germ equivalent at x in X if first S and T are germ equivalent at x, say, and then moreover, for some smaller neighbourhood V with.

Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x.

If and have additional structure, it is possible to define subsets of the set of all maps from X to Y or more generally sub-presheaves of a given presheaf and corresponding germs: some notable examples follow.

Anyone who tried to remedy some of the most glaring defects in our form of democracy was denounced as a traitorous red whose real purpose was the destruction of our government.

Research, on the other hand, has shown many stepmothers to be eminently successful, some far better than the real mothers.

`` I've served as a counsel for the U.N. for some years, specializing particularly in real estate matters or other problems that the regular U.N. legal staff might not be equipped to handle.

But the real beginnings of this development in him go back to the opposing of grammar school, and probably if it had not been this occasion and these Latin lines it would have been some others, such as the first prolusion, that set off this streak in him of unbridled and scathing verbal attack on an enemy.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

But these are dreamed in original action, in some particular continuity which we don't remember having seen in real life.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

It follows, then, provided the possibilities have been exhausted, that the only real alternative is the general viewpoint of the `` left '', which has been represented on the Continent by Fritz Buri and, to some extent at least, is found in much that is significant in American and English theology.

They would become tagged as men not interested in being purely real estate `` professionals '' but agitators for some kind of `` cause '' or `` reform '', and this was no longer to be a `` pro ''.

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

While most everyone has an experience with anxiety at some point in their lives, as it is a common reaction to real or perceived threats of all kinds, most do not develop long-term problems with anxiety.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

Similarly, a character can find any possible universe, but they can spend character points to know of or inhabit shadows which are ( in some sense ) " real " and therefore useful.

This is regarded by some historians as the real end of the Byzantine Empire.

This has led to some debate over how real, and how universal, phonemes really are ( see phoneme for details ).

Although they are generally classified as a cappella metal, the band also includes a drummer, and use amplifiers on some songs to distort the voice to sound more like a real guitar.

The real occasion of the outbreak of the war was the refusal of Athens to restore the hostages some twenty years later.

Notwithstanding the new Amir's incapacity, and some jealousy between the real leaders, Abdur Rahman and his uncle, they again routed Sher Ali's forces, and occupied Kandahar in 1867.

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.

Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line.

Peach or plum flavoring appeared, then more fruit flavors were added until, in some variations, the tea was removed entirely in favor of real fruit.