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Compactification (mathematics)

In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space; the methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape". Consider the real line with its ordinary topology; this space is not compact. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞; the resulting compactification can be thought of as a circle. Every sequence that ran off to infinity in the real line will converge to ∞ in this compactification. Intuitively, the process can be pictured as follows: first shrink the real line to the open interval on the x-axis; this point is our new point ∞ "at infinity". A bit more formally: we represent a point on the unit circle by its angle, in radians, going from -π to π for simplicity. Identify each such point θ on the circle with the corresponding point on the real line tan.

This function is undefined at the point π. Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without ∞. What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below, it is possible to compactify the real line by adding two points, +∞ and -∞. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X, it is useful to embed topological spaces in compact spaces, because of the special properties compact spaces have. Embeddings into compact Hausdorff spaces may be of particular interest. Since every compact Hausdorff space is a Tychonoff space, every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is true; 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.

For any topological space X the one-point compactification αX of X is obtained by adding one extra point ∞ and defining the open sets of the new space to be the open sets of X together with the sets of the form G ∪, where G is an open subset of X such that X \ G is compact. The one-point compactification of X is only if X is Hausdorff and locally compact. Of particular interest are Hausdorff compactifications, i.e. compactifications in which the compact space is Hausdorff. A topological space has a Hausdorff compactification if and only. In this case, there is a unique "most general" Hausdorff compactification, the Stone–Čech compactification of X, denoted by βX. "Most general" or formally "reflective" means that the space βX is characterized by the universal property that any continuous function from X to a compact Hausdorff space K can be extended to a continuous function from βX to K in a unique way. More explicitly, βX is a compact Hausdorff space containing X such that the induced topology on X by βX is the same as the given topology on X, for any continuous map f:X → K, where K is a compact Hausdorff space, there is a unique continuous map g:βX → K for which g restricted to X is identically f.

The Stone–Čech compactification can be constructed explicitly as follows: let C be the set of continuous functions from X to the closed interval. Each point in X can be identified with an evaluation function on C, thus X can be identified with a subset of the space of all functions from C to. Since the latter is compact by Tychonoff's theorem, the closure of X as a subset of that space will be compact; this is the Stone–Čech compactification. Walter Benz and Isaak Yaglom have shown how stereographic projection onto a single-sheet hyperboloid can be used to provide a compactification for split complex numbers. In fact, the hyperboloid is part of a quadric in real projective four-space; the method is similar to that used to provide a base manifold for group action of the conformal group of spacetime. Real projective space RPn is a compactification of Euclidean space Rn. For each possible "direction" in which points in Rn can "escape", one new point at infinity is added; the Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP1.

Note however that the projective plane RP2 is not the one-point compactification of the plane R2 since more than one point is added. Complex projective space CPn is a compactification of Cn. Passing to projective space is a common tool in alge

Flor Isava Fonseca

Flor Isava Fonseca is a Venezuelan sportswoman and writer as well as prominent member of Venezuelan society. For a number of years, she was the vice president of the Venezuelan Red Cross, following in the footsteps of her mother, appointed president of this institution years before, she is an active member of the Venezuelan associated press committee and has devoted much her time and effort in helping the poor, the blind and prisoners from all over the country. She was twice married, first to Venezuelan media personality Luis Teófilo Núñez Arismendi, with whom she had three children and afterwards to Domingo Lucca Romero. In 1981 Isava-Fonseca and Finnish Pirjo Häggman were the first women to be elected to the International Olympic Committee, she was the first woman to serve on the executive board in 1990. IOC member profile

Jean-Baptiste Nouvion

Jean-Baptiste Nouvion was a French prefect and a colonial administrator in Algeria. He made the success of the French aperitif Sirop de Picon, he started his career as chief of staff of the Civil governor of Algeria, Gustave Mercier Lacombe from 1859 to 1861. After several positions as sub prefect in Saint Nazaire and Philippeville, he became the prefect of Oran in Algeria from 1873 until 1879. In 1862 the French government invited industry to take part in the Universal Exhibition in London. Jean-Baptiste Nouvion, the sub-prefect of Philippeville, urged Gaëtan Picon to bring his aperitif Sirop de Picon to the exhibition. But, failing to convince the manufacturer to take part, the sub-prefect stubbornly took it upon himself to ship a case of African Amer to London; the product ended up crowned with a bronze medal in the bitter aperitif category adding to Gaëtan Picon's eventual fortune. A city near Oran was named Nouvion as a recognition of his work. After the Independence of Algeria, the name of the city was changed to El Ghomri.

· France: Officier of the Legion of Honour by the emperor Napoléon III in 1865 Spain: Commandeur of the Order of Isabella the Catholic Tunisia: Grand officier of the Order of Glory

Moro reflex

The Moro reflex is an infantile reflex that develops between 28–32 weeks of gestation and disappears between 3–6 months of age. It is a response to a sudden loss of support and involves three distinct components: spreading out the arms pulling the arms in crying It is distinct from the startle reflex. Unlike the startle reflex, the Moro reflex does not decrease with repeated stimulation; the primary significance of the Moro reflex is in evaluating integration of the central nervous system. Ernst Moro elicited the Moro reflex by slapping the pillow on both sides of the infant's head. Other methods have been used since including lowering the infant to a sudden stop and pinching the skin of the abdomen. Today, the most common method is the head drop, where the infant is supported in both hands and tilted so the head is a few centimeters lower than the level of the body; the Moro reflex may be observed in incomplete form in premature birth after the 25th week of gestation, is present in complete form by week 30.

Absence or asymmetry of either abduction or adduction by 2 to 3 months age can be regarded as abnormal, as can persistence of the reflex in infants older than 6 months. Furthermore, absence during the neonatal period may warrant assessment for the possibility of developmental complications such as birth injury or interference with brain formation. Asymmetry of the Moro reflex is useful to note, as it is always a feature of root, plexus, or nerve disease; the Moro reflex is absent in infants with kernicterus. An exaggerated Moro reflex can be seen in infants with severe brain damage that occurred in-utero, including microcephaly and hydrancephaly. Exaggeration of the Moro reflex, either due to low threshold or excessive clutching occurs in newborns with moderate hypoxic-ischemic encephalopathy; the Moro reflex is exaggerated in infants withdrawing from narcotics. Persistence of the Moro reflex beyond 6 months of age is noted only in infants with severe neurological defects, including cerebral palsy.

The Moro reflex was first described in western medicine by Austrian pediatrician Ernst Moro in 1918. Moro referred to it as the Umklammerungsreflex. In this publication, he stated: "When a young infant is placed on the examination table and one taps with hands on both sides of the pillow, there follows a rapid symmetrical extending abduction of both extremities, which approach each other in adduction thereafter". According to him, this reflex should disappear after the infant's first 3–6 months of life. Since the moro reflex has been used to detect early neurological problems in infants. Absence or prolonged retention of Moro reflex can be signs that the infants need neurological attention; the Moro reflex may be a survival instinct to help the infant cling to its mother. If the infant lost its balance, the reflex caused the infant to embrace its mother and regain its hold on the mother’s body. Medline Plus: Moro reflex Video of the Moro Reflex from the University of Utah Pathologic Moro Reflex in an adult following acute demyelinating lesion of unknown origin in the medulla oblongata

Stephanie Perkins

Stephanie Perkins is an American author, known for her books Anna and the French Kiss and the Boy Next Door, The New York Times bestseller Isla and the Happily Ever After and There's Someone Inside Your House, the latter of which will be adapted into a film of the same name and will be released on Netflix. Perkins was born in South Carolina. During her formative years, she lived in Arizona with her family, attended universities in California and Georgia. After spending a year living in San Francisco, she moved away to live with her husband, Jarrod Perkins in Asheville, North Carolina. Anna and the French Kiss Lola and the Boy Next Door Isla and the Happily Ever After There's Someone Inside Your House My True Love Gave To Me: Twelve Holiday Stories Summer Days and Summer Nights: Twelve Love Stories Listen Up Award YALSA's Best Fiction for Young Adults Official page on Blogger

Thomas Duncombe (died 1746)

Thomas Duncombe of Duncombe Park, Yorks was a British Tory politician who sat in the House of Commons in two parliaments between 1711 and 1741 Duncombe was born Thomas Browne, the only son of Thomas Browne, merchant, of St Margaret’s, Westminster and his wife Ursula Duncombe, daughter of Alexander Duncombe of Drayton, Buckinghamshire. His father was involved with his uncle, the banker Sir Charles Duncombe, in making government loans in the reign of Charles II and continued to do so on his own after 1690, lending various sums.. He matriculated at Christ Church, Oxford on 27 April 1703, aged 19 and was admitted at Inner Temple in 1709. In 1711 he succeeded to the Yorkshire estates, of his uncle Sir Charles Duncombe and assumed the name of Duncombe, he married Sarah Slingsby, daughter of Sir Thomas Slingsby, 4th Baronet, of Scriven, Yorkshire on 18 August 1714. Duncombe was returned unopposed as Member of Parliament for Downton in a by-election on 9 May 1711 in succession to his late uncle, he was acting as a stop-gap for the rest of that Parliament and was a Tory like the rest of his family.

He is not recorded as having voted in any of the divisions during his time in Parliament but it is not possible to distinguish his contributions from those of other Duncombes. He did not stand for Parliament at the 1713 British general election. Duncombe succeeded to the estates of his father in 1720, he was High Sheriff of Yorkshire for the year 1727 to 1728. At the 1734 British general election, he was returned as MP for Ripon by John Aislabie, whose daughter had married his brother-in-law, Sir Henry Slingsby, 5th Baronet, he voted against the Administration, was not asked to stand at the 1741 British general election, Duncombe died on 23 March 1746 leaving three sons and two daughters. His eldest son, Thomas became a British politician