The collective form is therefore similar in many respects to an English mass noun like " rice ", which in fact refers to a collection of items which are logically countable.

The last fact comes from the fact that is compact Hausdorff, and hence ( since compact metrisable spaces are necessarily second countable ); as well as the fact that compact Hausdorff spaces are metrisable exactly in case they are second countable.

The collective form is therefore similar in many respects to an English mass noun such as " rice ", which in fact refers to a collection of items which are logically countable.

:: This follows from the fact that a countable union of countable sets is countable.

For example, the index of H in G may be countable or uncountable, depending on whether H has a countable number of cosets in G. Note that the index of H is at most the order of G, which is realized for the trivial subgroup, or in fact any subgroup H of infinite cardinality less than that of G.

( One can prove that the covering space is second-countable from the fact that the fundamental group of a manifold is always countable.

( This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.

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.

The fact that these countable models of ZF still satisfy the theorem that there are uncountable sets is not considered a pathology ; van Heijenoort ( 1967 ) describes it as " a novel and unexpected feature of formal systems.

A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family

However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence.

On reaching Sanlucar he was detained again, the authorities having discovered a shortfall in his complement of men and horses, and the fact that large numbers of his crew were not Spanish.

Subcultures can exist at all levels of organizations, highlighting the fact that there are multiple cultures or value combinations usually evident in any one organization that can complement but also compete with the overall organisational culture.

When the element is a particle, it can not ( or no longer ) be construed as a preposition, but rather it is a particle by virtue of the fact that it does not take a complement.

There are, in fact, two ways to measure the spherical area enclosed by a simple closed curve, but the inequality is symmetric with the respect to taking the complement.

This is evident from the fact that although Lin's martial arts seem to counter Wang's, they actually complement his skills.

It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a union of open intervals ; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.

: From the point of view of the definition, this is an expression of the fact that the knot complement is a homology circle, generated by the covering transformation.

In fact, if is a fiber bundle where is the knot complement, let represent the monodromy, then where is the induced map on homology.

In fact this original work was a new interpretation on the history of the ideas of all time — a complement to Thomas Kuhn's work or even a more accurate look at the dynamics of invention.

The fact that back seat passengers could only open their windows down to approximately a third of their depth before further opening was blocked by the presence of the wheel arches was held out as a safety feature to complement the fitting of child-proof locks, given that back-seat passengers would no doubt include small children.

The first is that Wehler credits leaders such as Admiral Alfred von Tirpitz and Prince Bernhard von Bülow with a greater degree of vision than they in fact possessed The second is that many of the right-wing pressure groups who advocated an imperialist policy for Germany were not government creations, and in fact often demanded far more aggressive policies then the government was willing to undertake The third was that many of the groups advocating imperialism demanded a policy of political and social reform at home to complement imperialism abroad Eley argued that what is required in thinking about social imperialism is a broader picture with an interaction between above and below, and a wider view of the relationship between imperialism abroad and domestic politics

Some of the most surprising complexity results shown to date showed that the complexity classes NL and SL are in fact closed under complement, whereas before it was widely believed they were not ( see Immerman-Szelepcsényi theorem ).

In fact, Erik Brynjolfsson and his colleagues found a significant positive relationship between IT investments and productivity, at least when these investments were made to complement organizational changes.

" In fact, he went on to say that Burkitt's Exposition was the inspiration for his own commentary on the Old Testament, to complement Burkitt's work on the New Testament.

The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.

In fact, a bounded linear functional on C < sub > c </ sub >( X ) need not remain so if the locally convex topology on C < sub > c </ sub >( X ) is replaced by the supremum norm, the norm of C < sub > 0 </ sub >( X ).

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure.

Completely regular spaces can be characterized by the fact that their topology is completely determined by C ( X ) or C *( X ).

An important fact about the weak * topology is the Banach – Alaoglu theorem: if X is normed, then the closed unit ball in X * is weak *- compact ( more generally, the polar in X * of a neighborhood of 0 in X is weak *- compact ).

Spec ( R ) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology.

In fact, the collection of all left cosets ( or right cosets ) of C in G is equal to the collection of all components of G. Therefore, the quotient topology induced by the quotient map from G to G / C is totally disconnected.

In fact, given a set X and a collection F of subsets of X that has these properties, then F will be the collection of closed sets for a unique topology on X.

None of these techniques, however, makes any difference to the basic principles of EIGRP, which exchanges a vector of distances to each known destination network without full knowledge of the network topology, and, as a matter of fact, similar techniques have been used in other distance-vector protocols ( notably DSDV, AODV and Babel ).

In fact they are a base for the standard topology on the real numbers.

In fact, any open sets in the space generated by a base may be safely added to the base without changing the topology.

This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives the same topology as one would get by taking formal power series in all inderteminates at once.

It does however converge in the usual topology of R, and in fact to the coefficient of exp ( X ).

The theorem crucially depends upon the precise definitions of compactness and of the product topology ; in fact, Tychonoff's 1935 paper defines the product topology for the first time.

Cisco literature refers to EIGRP as a " hybrid " protocol, despite the fact it distributes routing tables instead of topology maps.

In fact, the Zariski topology is the weakest topology ( with the fewest open sets ) in which this is true and in which points are closed.

Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra ; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.

In fact, an arbitrary topology on X satisfies the T < sub > 1 </ sub > axiom if and only if it contains the cofinite topology.

In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other.