An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset.

A simple example of a space which is not separable is a discrete space of uncountable cardinality.

For example the algebraic closure, the separable closure, the cyclic closure et cetera.

As an example, a complete separable metric space X can be described by a sequence of real numbers, and therefore by a sequence of integers, simply by listing the distances between each pair of elements of a given countable dense set.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of p, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then is a resolvent for the alternating group.

For example, the separable verb anfangen (" to start ") consists of the separable prefix an and the root fangen:

Many English phrasal verbs are separable, in the sense that if they are transitive then the object is placed between the core verb and the particle if the object is a pronoun ( and optionally if it is a short noun phrase, but not if it is a long noun phrase as in the example above ).

This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.

For example, one of the key questions in the United States v. Microsoft antitrust trial was whether Microsoft's Internet Explorer web browser was part of its Windows operating system or a separable piece of application software.

Though algebras of operators are studied in various contexts ( for example, algebras of pseudo-differential operators acting on spaces of distributions ), the term operator algebra is usually used in reference to algebras of bounded operators on a Banach space or, even more specially in reference to algebras of operators on a separable Hilbert space, endowed with the operator norm topology.

For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable.

A separable channel is an example of local operation and classical communication ( LOCC ).

John Locke, for example, defines body simply as ".. something that is solid and extended, whose parts are separable and movable different ways.

( Furthermore an example of a separable algebra extension since is a separability element satisfying ea = ae for all a in A as well as.

For example, even the most successful executive lives in a two-room apartment while ordinary people rent space in the stairwells of office buildings in which to sleep at night ; ;

The spaciousness of the Tudor cooking areas, for example, will provide needed space for the extra television sets required by modern butlers, cooks and maids.

Marco Polo's systematic observations of nature, anthropology, and geography are another example of studying human variation across space.

The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

For example, consider the sample space

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Exile ( 1988 video game ) | Exile is an example of a game where the developers left non-graphical data visible in the display buffer to gain additional memory space.

Animals can respond to extreme heat, for example, through natural heat acclimation or by burrowing into the ground to find a cooler space.

This accompanied or facilitated other important evolutionary developments: the bilaterian body plan ; the coelom, an internal cavity that provided space for a circulatory system and, in some animals, formed a hydrostatic skeleton which enables worm-like animals to burrow ; metamerism, in which the body was built of repeated " modules " which could later specialize, for example the heads of most arthropods are composed of fused, specialized segments.

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

Similarly, as an infinite-dimensional example, the Lebesgue space L < sup > p </ sup > is always a Banach space but is only a Hilbert space when p = 2.

Note that if we regard the product as a vector space, then B is not a linear transformation of vector spaces ( unless or ) because, for example.

Some early ' skyscrapers ' were made in masonry, and demonstrated the limitations of the material – for example, the Monadnock Building in Chicago ( opened in 1896 ) is masonry and just 17 stories high ; the ground walls are almost thick, clearly building any higher would lead to excessive loss of internal floor space on the lower floors.

For example, in three-dimensional complex Euclidean space,

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

For example, a shopping mall does not provide the merchandise a shopper is seeking, but provides space and services for retailers that serve the shopper.

An example of a compact space is the unit interval of real numbers.

The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself: an open ( or half-open ) interval of the real numbers is not compact.

For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.

Not every countably compact space is compact ; an example is given by the first uncountable ordinal with the order topology.

Not every compact space is sequentially compact ; an example is given by 2 < sup ></ sup >, with the product topology.

for example, the mode of bravery to this anonymous folk poem: `` They brought me news that Spring is in the plains And Ahmad's blood the crimson tulip stains ; ;

For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.

This almost trivial example is nevertheless suggestive, for there are some elements in common between the antique fear that the days would get shorter and shorter and our present fear of war.

Perhaps the most illuminating example of the reduction of fear through understanding is derived from our increased knowledge of the nature of disease.

Beckett's own work is an example.

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

His name is Praisegod Piepsam, and he is rather fully described as to his clothing and physiognomy in a way which relates him to a sinister type in the author's repertory -- he is a forerunner of those enigmatic strangers in `` Death In Venice '', for example, who represent some combination of cadaver, exotic, and psychopomp.

Gustaf Vasa is a superb example, and Charles 10,, the conqueror of Denmark, hardly less so.

For example, suppose a man wearing a $200 watch, driving a 1959 Rolls Royce, stops to ask a man on the sidewalk, `` What time is it ''??

In the extreme and oversimplified example suggested in Figure 3, the organization is more easily understood and more predictable in behavior.

The assumptions upon which the example shown in Figure 3 is based are: ( A ) One man can direct about six subordinates if the subordinates are chosen carefully so that they do not need too much personal coaching, indoctrinating, etc..

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

In the calm which follows the reading of a poem, for example, is the effect produced by the enforced quiet, by the musical quality of words and rhythm, by the sentiments or sense of the poem, by the associations with earlier readings, if it is familiar, by the boost to the self-esteem for the semi-literate, by the diversion of attention, by the sense of security in a legitimized withdrawal, by a kind license for some variety of fantasy life regarded as forbidden, or by half-conscious ideas about the magical power of words??

English philosopher Samuel Alexander's debt to Wordsworth and Meredith is a recent interesting example, as also A. N. Whitehead's understanding of the English romantics, chiefly Shelley and Wordsworth.

In his book Civilization And Ethics Albert Schweitzer faces the moral problems which arise when moral law is recognized in business life, for example.

Easily the best known of these three novels is The Space Merchants, a good example of a science-fiction dystopia which extrapolates much more than the impact of science on human life, though its most important warning is in this area, namely as to the use to which discoveries in the behavioral sciences may be put.

And to do this requires first of all the kind of information about people which is provided by the scientists in industrial anthropology and consumer research, who, for example, tell Courtenay that three days is the `` optimum priming period for a closed social circuit to be triggered with a catalytic cue-phrase '' -- which means that an effective propaganda technique is to send an idea into circulation and then three days later reinforce or undermine it.

One specific example is a secret `` fraternity '' which will `` coordinate anti-Communist efforts ''.