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first and countable
* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.
This space is Hausdorff, paracompact and first countable.
A countable product of second countable spaces is second countable, but an uncountable product of second countable spaces need not even be first countable.
Then X has cardinality at most and cardinality at most if it is first countable.
However, the product of uncountably many metrizable spaces, each with at least two points, fails to be first countable.
In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite ( but critically important ) variation of the Dedekind cut.
Fréchet spaces can be defined in two equivalent ways: the first employs a translation-invariant metric, the second a countable family of semi-norms.
* Every first countable space is sequential.
A-dimensional foliation of an-manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed-dimensional submanifolds ( the leaves of the foliation ) of, such that for every point in, there is a chart with homeomorphic to containing such that for every leaf, meets in either the empty set or a countable collection of subspaces whose images under in are-dimensional affine subspaces whose first coordinates are constant.
Beyond the countable, the first uncountable ordinal is usually denoted ω < sub > 1 </ sub >.
Also, ω < sub > 1 </ sub > is the smallest uncountable ordinal ( to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers ; each such well-ordering defines a countable ordinal, and ω < sub > 1 </ sub > is the order type of that set ), ω < sub > 2 </ sub > is the smallest ordinal whose cardinality is greater than ℵ < sub > 1 </ sub >, and so on, and ω < sub > ω </ sub > is the limit of ω < sub > n </ sub > for natural numbers n ( any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the ω < sub > n </ sub >).
In 1920, he greatly simplified the proof of a theorem Leopold Löwenheim first proved in 1915, resulting in the Löwenheim – Skolem theorem, which states that if a first-order theory has an infinite model, then it has a countable model.
The nodes are thus in a one-to-one correspondence with finite ( possibly empty ) sequences of positive numbers, which are countable and can be placed in order first by sum of entries, and then by lexicographic order within a given sum ( only finitely many sequences sum to a given value, so all entries are reached – formally there are a finite number of compositions of a given natural number, specifically 2 < sup > n − 1 </ sup > compositions of n ≥ 1 ;), which gives a traversal.
However, X /~ is not first countable at the coset of the identified points and hence also not second countable.
Thus the seeming contradiction is that a model which is itself countable, and which contains only countable sets, satisfies the first order sentence that intuitively states " there are uncountable sets ".
In particular, this implies that if the first order versions of Zermelo's axioms of set theory are satisfiable, they are satisfiable in some countable model.
In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable.

first and separable
A separable linear ordinary differential equation of the first order
Together, these first two examples give a different proof that n-dimensional Euclidean space is separable.
* The first uncountable ordinal ω < sub > 1 </ sub > in its order topology is not separable.
In 1690, Jacob Bernoulli became the first person to develop the technique for solving separable differential equations.
One may then define an index of separability,, which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.
"... a remarkable result of this first period is his topological embedding of separable metric spaces into compact spaces of the same ( finite ) dimension.
An alternative approach, equivalent to the one given here, is first to define " Polish metric space " to mean " complete separable metric space ", and then to define a " Polish space " as the topological space obtained from a Polish metric space by forgetting the metric.
A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
The two power cars were articulated together with a span bolster, giving them an effective wheel arrangement of B-B + B-B, but they were not articulated with the train and therefore formed the Union Pacific's first separable diesel locomotive.
The cap ( pileus ) of Psilocybe azurescens is 30 – 100 mm in diameter, conic to convex, expanding to broadly convex and eventually flattening with age with a pronounced, persistent broad umbo ; surface smooth, viscous when moist, covered by a separable gelatinous pellicle ; chestnut to ochraceous brown to caramel in color often becoming pitted with dark blue or bluish black zones, hygrophanous, fading to light straw color in drying, strongly bruising blue when damaged ; margin even, sometimes irregular and eroded at maturity, slightly incurved at first, soon decurved, flattening with maturity, translucent striate and often leaving a fibrillose annular zone in the upper regions of the stipe.

first and Hausdorff
It was first noted in print by Campbell ( 1897 ); elaborated by Henri Poincaré ( 1899 ) and Baker ( 1902 ); and systematized geometrically, and linked to the Jacobi identity by Hausdorff ( 1906 ).
They were first introduced by Kazimierz Kuratowski, in a slightly different form that applied only to Hausdorff spaces.
It seems that this distance was first introduced by Hausdorff in his book " Grundzüge der Mengenlehre " the first edition published in 1914.
The notion of Gromov – Hausdorff convergence was first used by Gromov to prove that
( Hausdorff first showed in the same paper the easier result that there is no countably additive measure defined on all subsets.
The separation axioms that were first studied together in this way were the axioms for accessible spaces, Hausdorff spaces, regular spaces, and normal spaces.

first and space
We note that, first, America has already made great contributions in the past two years to the world's fund of knowledge of astrophysics and space science.
But first, we must define two terms so that their meaning will be clearly understood: form -- any unique sequence of alphabetic characters that can appear in a language preceded and followed by a space ; ;
In their very first collages, Braque and Picasso draw or paint over and on the affixed paper or cloth, so that certain of the principal features of their subjects as depicted seem to thrust out into real, bas-relief space -- or to be about to do so -- while the rest of the subject remains imbedded in, or flat upon, the surface.
At the very first, then, Brumidi was required, by the classically pyramidal shape of his central group, to fill in the triangular space above the seated girl on Liberty's right, before starting on the allegorical figures themselves.
An early pioneer of radio astronomy was Grote Reber, an amateur astronomer who constructed the first purpose built radio telescope in the late 1930s to follow up on the discovery of radio wavelength emissions from space by Karl Jansky.
The 2600 port was the first game to use a bank-switched cartridge, doubling available ROM space.
* 1961 – The Russian ( Soviet ) cosmonaut Yuri Gagarin becomes the first human to travel into outer space and perform the first manned orbital flight, in Vostok 3KA-2 ( Vostok 1 ).
His report of life there covers a wide range of topics, such as marriage in heaven ( where all angels are married ), children in heaven ( where they are raised by angel parents ), time and space in heaven ( there are none ), the after-death awakening process in the World of Spirits ( a place halfway between Heaven and Hell and where people first wake up after death ), the allowance of a free will choice between Heaven or Hell ( as opposed to being sent to either one by God ), the eternity of Hell ( one could leave but would never want to ), and that all angels or devils were once people on earth.
While Renaissance artists sought nature to find their style, the Mannerists looked first for a style and found a manner. In Mannerist paintings, compositions can have no focal point, space can be ambiguous, figures can be characterized by an athletic bending and twisting with distortions, exaggerations, an elastic elongation of the limbs, bizarre posturing on one hand, graceful posturing on the other hand, and a rendering of the heads as uniformly small and oval.
First conceived during the Presidency of Dwight D. Eisenhower as a three-man spacecraft to follow the one-man Project Mercury which put the first Americans in space, Apollo was later dedicated to President John F. Kennedy's national goal of " landing a man on the Moon and returning him safely to the Earth " by the end of the 1960s, which he proposed in a May 25, 1961 address to Congress.
Using space exploration as a symbol of national prestige, he warned of a " missile gap " between the two nations, pledging to make the U. S. not " first but, first and, first if, but first period.
On April 12, 1961, Soviet cosmonaut Yuri Gagarin became the first person to fly in space, reinforcing American fears about being left behind in a technological competition with the Soviet Union.
* Albert I, the first monkey in outer space ( 1948 )
* 2001 – Millionaire Dennis Tito becomes the world's first space tourist.
* 1981 – STS-1 – The first operational space shuttle, Columbia ( OV-102 ) completes its first test flight.
Apollo 16 was the tenth manned mission in the United States Apollo space program, the fifth and penultimate to land on the Moon and the first to land in the lunar highlands.
NASA later confirmed the object to be a piece of the first stage of the Saturn V rocket that launched Apollo 16 into space.
The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.
* 1971 – Launch of Salyut 1, the first space station.
* 2003 – Yuri Malenchenko becomes the first person to marry in space.

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