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Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where for any two distinct points there exist neighbourhoods of each which are disjoint from each other. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most used and discussed, it implies the uniqueness of limits of sequences and filters. Hausdorff spaces are named after one of the founders of topology. Hausdorff's original definition of a topological space included the Hausdorff condition as an axiom. Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint. X is a Hausdorff space; this condition is the third separation axiom, why Hausdorff spaces are called T 2 spaces. The name separated space is used. A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods.

Preregular spaces are called R 1 spaces. The relationship between these two conditions is. A topological space is only if it is both preregular and Kolmogorov. A topological space is only if its Kolmogorov quotient is Hausdorff. For a topological space X, the following are equivalent: X is a Hausdorff space. Limits of nets in X are unique. Limits of filters on X are unique. Any singleton set; the diagonal Δ = is closed as a subset of the product space X × X. All spaces encountered in analysis are Hausdorff. More all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions. A simple example of a topology, T1 but is not Hausdorff is the cofinite topology defined on an infinite set. Pseudometric spaces are not Hausdorff, but they are preregular, their use in analysis is only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still at least preregular, they replace it with its Kolmogorov quotient, Hausdorff.

In contrast, non-preregular spaces are encountered much more in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff, in fact is neither; the related concept of Scott domain consists of non-preregular spaces. While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit. Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space. Hausdorff spaces are T1. Preregular spaces are R0. Another nice property of Hausdorff spaces is; this may fail in non-Hausdorff spaces such as the Sierpiński space.

The definition of a Hausdorff space says. It turns out that this implies something, stronger: in a Hausdorff space every pair of disjoint compact sets can be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint; this is an example of the general rule that compact sets behave like points. Compactness conditions together with preregularity imply stronger separation axioms. For example, any locally compact preregular space is regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers; the Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, every compact Hausdorff space is normal Hausdorff. The following results are some technical properties regarding maps to and from Hausdorff spaces. Let f: X → Y be a continuous function and suppose Y is Hausdorff.

The graph of f, { ( x, f

Mark Way

Wilfrid Lewis Mark Way was a British Anglican bishop. He was the third Bishop of Masasi during the third quarter of the 20th century. Born on 12 May 1905 and educated at Rossall School and Trinity College, Cambridge he was ordained in 1929, he was made a deacon in Advent 1928 and ordained a priest the next Advent — both times by Albert David, Bishop of Liverpool, at Liverpool Cathedral. After curacies at St Faith, Great Crosby and St Bartholomew, Brighton he became an UMCA Missionary in Tanzania, he was Curate of Korogwe Priest in charge of several Zanzibar Missions and Warden of Kalole Theological College. He was consecrated a bishop on St Luke's Day 1952 at Westminster Abbey. On his return to England he was Rector of Averham, during that time an Assistant Bishop of Southwell, he died on 30 July 1982

Tom Scavo

Thomas “Tom” Scavo is a fictional character on the ABC television series Desperate Housewives, played by actor Doug Savant. Tom was credited as a recurring role throughout Season 1, but became credited as a series regular in Season 2. Early storylines revolved around Tom's position at an advertising firm, causing him to travel leaving Lynette alone to run the family. Starting in Season 3, Tom's storyline turned to a focus of him trying to find what was next for him, due to being tired of working in advertising; when he decides, he opens Scavo's Pizzeria, much to Lynette's dismay. After the five year jump, it becomes apparent that Tom is now experiencing a mid-life crisis, as he purchased and restored an early model convertible. After Lynette gives birth to Paige, it is discovered that Tom is experiencing Male Post-Partum Depression Tom separates from Lynette at the end of Season 7 and dates Jane for the greater part of Season 8, however and Lynette reunite at the end of the series. Tom Scavo was the eldest of three children born to Rodney Scavo.

He had three relationships in his life: first was one-night-stand Nora Huntington, Annabel Foster when they worked together in Advertising and one-night-stand with Renee Perry, during the short time when breaking up of engagement with Lynette. Lynette Lindquist and Tom met in the same company, they moved to Wisteria Lane in the following year. Lynette has four children with Tom: twins Preston and Porter born in 1998, Parker in 1999, Penny in 2004, she had a successful career, but she gave up all that to become a stay-at-home mother. Tom was a working dad, always away from home, supporting financially his family. Tom Scavo is out of town and seems oblivious to Lynette's problems. Tom has been keeping a secret from his wife that only his father, Rodney Scavo, knew about his one-night-stand with Renee. Tom decides to become a house-husband after losing his promotion to Tim Douggan and subsequently quits his job, he seems to love Lynette, being prepared to discuss family matters with her if she feels that they need to.

Midway through the second season, Tom returns to the workplace, sharing an employer with his wife for the second time. When Lynette's boss asks Lynette to send instant messages to his wife to help him heat up his sex life, his wife found out and threatened to leave him unless he fires the person who sent the IMs. Feeling he cannot fire Lynette, he tells his wife it was Tom, deciding to fire him instead but Lynette warns him he will need reasonable cause. Much to Lynette's horror, Ed finds it - discovering; when Ed confronts him, the conversation ends with Tom punching Ed in the face. As grounds for dismissal have been met, double grounds in fact, Tom is fired. Tom is found out to be meeting another woman after Lynette follows him to Atlantic City and sees them embracing. Lynette finds out that this woman is the mother of his daughter - which he did not known he has - born long before he married Lynette, it is revealed that the woman is Nora Huntington, they had a one-night stand years before he had met Lynette.

She was claiming that she conceived his child, a daughter named Kayla Huntington, but never told him. Tom confesses to Lynette that he wants to open up a pizzeria. Nora, however undermines him behind his back and tries to get Lynette to tell him to drop the idea while planning to tell him that she'd support him, she wanted him back, determined to make a proper family for Kayla but Tom rejects her and tells Lynette, who warns Nora to keep her distance in future. Tom and Lynette plan to fight Nora for custody of Kayla but before they get to court, Nora confronts Lynette in the supermarket but gets caught in hostage crisis; when Lynette tells Carolyn that Nora made a move on Tom, Carolyn kills Nora. Lynette promises Nora just before she dies. Tom opens his pizzeria and with Lynette's help, gets it ready for the grand opening, overcoming a few hitches along the way, he tries organising a surprise for their 9th wedding anniversary but it goes wrong, leaving Lynette out in the woods at night and rushes to meet her.

A little while Tom's back is injured and he is confined to bed rest so Lynette hires a new chef and Mrs McCluskey to look after Tom and the children when she's at work. One night and the kids come to the pizzeria to surprise her, where Kayla sees chemistry between Lynette and Rick and tells Tom. So Tom suggests that they fire Rick but Lynette is not keen. After Lynette and Rick are locked in the freezer when the restaurant is robbed, they cuddle to keep warm but Tom sees the CCTV tape and confronts Rick, telling him that although his marriage to Lynette is shaky, it is rock solid and Rick would never break it up. Tom asks Rick to quit but he refuses, claiming Lynette "doesn't want him gone". Rick tells Lynette about his lunch with Tom and his feelings for her and tries to get her to admit her feelings for him but Lynette refuses and fires him. After finding out Tom was returning to work, Lynette sobbed in her bathroom distraught. Lynette and Tom’s marriage continues to decline and they don't speak for days.

Trying to rebuild their marriage, Tom gets his old college friend/marriage counselor to speak to them without Lynette knowing, but she realizes and becomes angrier about his'ambush therapy'. With Tom’s doctor informing him on his back recovery, Tom suggests to Lynette they either talk or have sex so Lynette agrees to sex but they become quite violent and Lynette falls off the bed, banging

Voidokilia beach

Voidokilia Beach is a popular beach in Messinia in the Mediterranean area. In the shape of the Greek letter omega, its sand forms a semicircular strip of dunes. On the land-facing side of the strip of dunes is Gialova Lagoon, an important bird habitat; the beach has been named "A Place of Particular Natural Beauty". It is part of a Natura 2000 protected area. Above the beach above this are the ruins of thirteenth-century Frankish castle. Overlooking the beach at the north eastern end is the tomb of Nestor's son, Thrasymedes of the Mycenaean period with Neolithic finds at the same site showing occupation as early as 4000 BC; the beach is presumed to be Homer's "sandy Pylos" where Telemachus was welcomed by King Nestor when searching for his father, Odysseus. According to myth, Nestor's Cave is. A hiking route begins from Voidokoilia; the climb towards Nestor's Cave begins at the south-west end of the beach, while after the cave the route continues towards Paliokastro. Voidokilia is next to Petrohori in Greece.

Though the ruins are not fenced off, the castle ruins are sign-posted as closed and tourists walk through the castle or around the walls at their own risk. Gialova's aquatic habitat, being an important stop for a variety of migratory birds, has been declared the southernmost Habitat of National Importance in the Balkans, it is considered a protected area hosting 258 species of birds out of which the 79 are included in the "Red Book". Parking for the beach is reached by sand tracks either from the Gialova Lagoon parking area or by following the Northside route from Petrochori; the beach is considered friendly for gay tourists. To the south of the beach stretches cape Koryphasion, where stood the classical city of Pylos. Nowadays few remains of the city are still extant, as in the 13th century Nicholas de Saint'Omer built a castle on the top of the cape using the extant building material consisting of the ancient stones. Beneath the castle a triangular opening on the rock, visible from Voidokoilia, marks the Cave of Nestor, according to mythology Hermes had hidden the stolen oxen of Helios, whereas according to Pausanias it was the place where were kept the cattle of the king Neleus.

The cave has been archaeologically investigated in various periods, starting from H Schliemann in 1874 and continuing with the research by Laucent and Kourouniotis in the beginning of the 20th century, C. Blegen and W Mc Donald, Sp. Marinatos in the 1950s and G. Korres and A. Samson in the 1980s, it has yielded finds dating from the Final Neolithic period down to the Late Helladic period as well as some classical and early Hellenistic traces. The dimensions of the cave are 20x16 with a heigh; the tomb of Thrasymedes, on the other hand, has been excavated by Sp. Marinatos in the 1950s and by G. S. Korres in the 1970s, since the latter noticed a pithos burial which had not been recorded by his predecessor; the tholos is 5 meters wide and its floor is covered with pebbles. It contained about 7 burials. Among the movable finds of the tomb count arrow heads, necklaces made of carnelian and amethyst, two Mycenaean pottery vessels etc; the first use of the tumulus dates to the Middle Helladic Period whereas the prime time of the tholos tomb was the Mycenaean period.

Around the tomb were found Late Classical and Hellenistic votive tablets as well as a small model of an altar, which points to some kind of hero or ancestors' cult. Voidokilia website information Information on Yialova birds Natura 2000

Florian W√ľnsche

Florian Wünsche is a German actor and well known for his role as Manuel Siewert in the children soap opera Schloss Einstein. He had his first small role in the feature movie Wer küsst schon einen Leguan? in 2003. This was followed by guest parts in two episodes of Ein Engel für alle and his first appearance in KRIMI. DE. After that he took the role of boarding school student Manuel Siewert in the popular teen soap Schloss Einstein in 2008. Wünsche lasted three seasons on the show, leaving Schloss Einstein in 2010, he took part in the second season of the Coming of age-series dasbloghaus.tv, before having another guest appearance in KRIMI. DE. In March 2011, Wünsche was cast as Emilio Sanchez in the soap opera Verbotene Liebe, he completed the show's Mallorca cast and joined the cast in Düsseldorf Germany. His first appearance was on 28 June 2011. After the show's cancellation in 2015 Florian Wünsche joined the cast of Stuttgart Homicide. In his role as Benedikt Förster he's part of the forensic team being a young coroner who has to deal with his male co-worker's jealousy as well as with his ex-girlfriend who tries to keep his daughter from him by allowing him to see her on tough conditions.

He starred in the popular German learning series, Nico's Weg where he and his friends adventure in Germany

Nigel Paulet, 18th Marquess of Winchester

Nigel George Paulet, 18th Marquess of Winchester is a British peer and the premier marquess of England. He succeeded a cousin in the title in 1968. Lord Winchester lives in South Africa. Nigel George Paulet was born on 23 December 1941 to Hazel Margaret Wheeler, he has two siblings, an older sister, Angela Jane Paulet, a younger brother, Timothy Guy Paulet. After his father's death in 1961, his mother remarried, to George Meyer, in 1962. Paulet's great-great grandfather was Lord Charles Paulet, the son of Charles Paulet, 13th Marquess of Winchester. Upon the death of his unmarried first cousin once removed, Richard Paulet, 17th Marquess of Winchester, on 5 March 1968, the 26-year-old Nigel became the 18th Marquess of Winchester, the highest-ranking marquess in the United Kingdom; the 17th Marquess' grandfather, Col. Charles William Paulet, was the great-grandfather of the 18th Marquess. On 25 November 1967, he married daughter of Major Aubrey John Hilton, they have two sons and one daughter: Christopher John Hilton Paulet, Earl of Wiltshire, heir apparent.