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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X; some related but stronger conditions are path connected connected, n-connected. Another related notion is locally connected, which neither follows from connectedness. A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology; some authors exclude the empty set as a connected space, but this article does not follow that practice. For a topological space X the following conditions are equivalent: X is connected, that is, it cannot be divided into two disjoint non-empty open sets.

X cannot be divided into two disjoint non-empty closed sets. The only subsets of X which are both open and closed are the empty set; the only subsets of X with empty boundary are the empty set. X cannot be written as the union of two non-empty separated sets. All continuous functions from X to are constant, where is the two-point space endowed with the discrete topology; the maximal connected subsets of a non-empty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, non-empty, their union is the whole space; every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is an open subset. However, if their number is infinite, this might not be the case. Let Γ x be the connected component of x in a topological space X, Γ x ′ be the intersection of all clopen sets containing x Then Γ x ⊂ Γ x ′ where the equality holds if X is compact Hausdorff or locally connected.

A space in which all components are one-point sets is called disconnected. Related to this property, a space X is called separated if, for any two distinct elements x and y of X, there exist disjoint open sets U containing x and V containing y such that X is the union of U and V. Clearly, any separated space is disconnected, but the converse does not hold. For example take two copies of the rational numbers Q, identify them at every point except zero; the resulting space, with the quotient topology, is disconnected. However, by considering the two copies of zero, one sees that the space is not separated. In fact, it is not Hausdorff, the condition of being separated is stronger than the condition of being Hausdorff; the closed interval in the standard subspace topology is connected. The union of is disconnected. ∪ is disconnected. A convex subset of Rn is connected. A Euclidean plane excluding the origin, is connected, but is not connected; the three-dimensional Euclidean space without the origin is connected, simply connected.

In contrast, the one-dimensional Euclidean space without the origin is not connected. A Euclidean plane with a straight line removed is not connected since it consists of two half-planes. ℝ, The space of real numbers with the usual topology, is connected. If a single point is removed from ℝ, the remainder is disconnected. However, if a countable infinity of points are removed from R n, where n ≥ 2, the remainder is connected. If n ≥ 3 R n remains connected after removal of countable many points. Any topological vector space, e.g. any Hilbert space or Banach space, over a connected field, is connected. Every discrete topological space with at least two elements is disconnected, in fact such a space is disconnected; the simplest example is the discrete two-point space. On the other hand, a finite set might be connected. For example, the spectrum of a discrete valuation ring is connected, it is an example of a Sierpiński space. The Cantor set is disconnected. If a space X is homotopy equivalent to a connected space X is itself connected.

The topologist's sine curve is an example of a set, connected but is neither path connected nor locally connected. The general linear group GL ⁡ consists of two connected components: the one with matrices of

Assistant Secretary of Defense for Legislative Affairs

The Assistant Secretary of Defense for Legislative Affairs, or ASD, is the head of the Office of the Secretary of Defense for Legislative Affairs, responsible for providing support to the Secretary of Defense in his/her dealings with the United States Congress. In addition to serving as SecDef's legislative adviser, the ASD promotes the Department of Defense's strategy, legislative priorities and budget before Congress. In carrying out these responsibilities, the ASD directs a team of managers, action officers, support personnel who help direct and manage communications and activities between Congress and elements of the Department of Defense; the ASD is considered a part of the Office of the Secretary of Defense. This office was established as Special Assistant upon the creation of the National Military Establishment in 1947; this was one of three special assistants to the first Secretary of Defense. The post was retitled Assistant Secretary of Defense in August 1949 based on amendments to the National Security Act that authorized three Assistant Secretaries of Defense.

The position was abolished in 1953, with its functions divided and transferred to the General Counsel and the Assistant Secretary of Defense, new posts established as the result of DoD Reorganization Plan No. 6 and Defense Directive 5122.1. This position was abolished again in 1957, with its functions divided and transferred to Assistant Secretary of Defense and Assistant to the Secretary of Defense, new posts established by Defense Directive 5105.13. Since 1957, the responsibilities of this position have stayed constant, but the title has changed between Assistant Secretary of Defense and Assistant to the Secretary of Defense five times because the Secretary of Defense has been authorized a limited number of assistant secretaries; the position was given statutory standing as the Assistant Secretary of Defense by the National Defense Authorization Act for FY1994, passed November 30, 1993. The table below includes both the various titles of this post over time, as well as all the holders of those offices.

The annual budget for the ASD is contained in the OSD's budget, under the Defense-Wide Operation and Maintenance account. The Obama administration cut funding for this position by over 37% in FY12

Toi Derricotte

Toi Derricotte is an American poet and a professor of writing at teh University of Pittsburgh. She won a 2012 PEN/Voelcker Award for Poetry. With Cornelius Eady, she co-founded Cave Canem Foundation, a summer workshop for African-American poets, she is the 2020 recipient of the Frost Medal for distinguished lifetime achievement in poetry awarded by the Poetry Society of America. Derricotte was born in Hamtramck, the daughter of Antonia Baquet, a Creole from Louisiana, Benjamin Sweeney Webster, a Kentucky native, half-sister to Benjamin, Jr. At around 10 or 11 years of age, she began a secret journal that included, among other things, the disintegration of her parents' marriage and the death of her grandmother on whom she was emotionally dependent. During her years at Detroit's Girls Catholic Central High School, Derricotte recounts a religious education that she felt was steeped in images of death and punishment, a Catholicism that, according to the poet, morbidly paraded "the crucifixion, martyrs in the Old Testament and the prayers of the Mass."

Coupled with these images were Derricotte's surreal reminiscences of childhood visits to her paternal grandparents' home, the bottom part of which served as a funeral parlor where bodies were prepared for viewing. She would stay overnight at her grandparents', unafraid, she would "pray over the bodies … … disturbed when young people died, babies." Her first attempt at sharing her poems with others came when, at fifteen, she visited a cousin, a medical school student, taking an embryology class. Encouraged by a trip they took to the Chicago Museum to see fetuses and embryos at various stages of development, careful not to show her poems to her parents who never "even alluded to babies before birth... talked to about sex," anxiously showed them to this cousin who pronounced them "sick, morbid." Faced with this unexpected rebuff, Derricotte remembers being faced with several choices: "I could have said something is wrong with me and stopped writing, or I could have continued to write, but written about the things I knew would be acceptable, or I could go back underground."

For Derricotte, the choice was obvious: rather than risk ostracism for writing about the forbidden, she opted "to go back underground." In 1959, Derricotte graduated from Girls Catholic Central and enrolled that autumn in Wayne State University as a special education major. In 1962, her junior year at Wayne State, she gave birth to a son in a home for unwed mothers; this act of rebellion was but a presage of things to come, as Derricotte, after graduating in 1965, left Detroit for the East Coast. At Wayne State University she earned a B. A. in 1965 and an M. A. in 1984 at New York University in English literature. Her move to New York City in 1967 was a momentous one, for it was here among white female intellectuals that Derricotte's poetic voice resurfaced. Unlike the African-American poets of the Black Arts Movement, many of whom heeded Amiri Baraka's call for an artistic expression, decidedly black nationalist and accessible, Derricotte wrote, instead personal, troubling difficult poems that talked more of black families haunted by gender oppression and familial strife than of Black Power and racial solidarity.

Having "paid her dues" as a student in numerous workshops where she endured the canon's litany of dead and near-dead white male poets such as Matthew Arnold, Ezra Pound, T. S. Eliot, Robert Lowell as the only black student, Derricotte first published in a "major" magazine, the New York Quarterly, in the fall of 1972, her literary reputation and publications flourished, culminating in her first book, The Empress of the Death House, published in 1978 by Lotus Press. Her second book, Natural Birth, was published in 1983 by The Crossing Press, her third book, first published in 1989 by University of Pittsburgh Press, has enjoyed second and third printings. In 1996, Norton Publishing Company accepted for publication Derricotte's The Black Notebooks, a book she began in 1974 when her family became one of the first black families to move into Upper Montclair, New Jersey, her work appears in Triquarterly, The Drunken Boat. In Derricotte's poetry, the taboo, the restricted, the repressed figure prominently.

Stylistically compared to so-called confessional poets like Sylvia Plath and Anne Sexton, Derricotte, in opting for candor over decorum, wants her "work to be a wedge into the world, as what is real and not what people want to hear." This self-dubbed "white-appearing Black person," reared as a Catholic in a black, working-class Detroit community, complicates the myth of monolithic blackness with poems that speak into consciousness obscure, unconventional black bodies. And in an academy whose poststructuralist theories either depersonalize bodies with esoteric discourse or overemphasize them with hyperbolic identity politics, Toi Derricotte's poems brave the charged, murky depths of much current poetry, stamping the language with her own complex, quirky vision. In 2012, Derricotte was elected a Chancellor of the Academy of American Poets, she is a professor of English at the University of Pittsburgh. In October 2019 "I": New and Selected Poems was named a finalist for the National Book Award for Poetry.

CollectionsDerricotte, Toi. The empress of the death house. Detroit: Lotus Press. Natural Birth, Ann Arbor: Firebrand Books, 1983, ISBN 9781563411205 Captivity, Pittsburgh: University of Pittsburgh Press, 1989, ISBN 9780822936282 Tender. University of Pittsburgh Press. 15 September 1997. ISBN 978-0-8229-7852-7; the Unde

St. Andrew's Presbyterian Church (Lunenburg)

St. Andrew's Presbyterian Church is a church in Lunenburg, Nova Scotia; the congregation is the longest history of any Presbyterian congregation in Canada. After meeting at a private house, the congregation worshipped in St. John's Anglican Church; the first church was built in 1770 and the first minister was Reverend Bruin Romkes Comingo, who served the community for 50 years until he died at age 95. The current church was built in the neo-gothic style and dates from 1828. Presbyterian Church in Canada Little Dutch Church St. John's Anglican Church Zion Evangelical Lutheran Church A sermon preached at Halifax, July 3d, 1770, at the ordination of the Rev. Bruin Romcas Camingoe to the Dutch Calvinistic Presbyterian Congregation at Lunenburg: by John Seccombe, of Chester, A. M. being the first preached in the province of Nova-Scotia, on such an occasion.

Lily Poulett-Harris

Lily Poulett-Harris was an Australian sportswoman and educationalist, notable for being the founder and captain of the first Women's cricket team in Australia. Poulett-Harris continued to play until forced to retire due to ill health from the tuberculosis, to claim her life. Born Harriet Lily Poulett-Harris on 2 September 1873, she was the youngest daughter of Richard Deodatus Poulett-Harris and his second wife, Elizabeth Eleanor, her father was renowned for being the head of the Hobart Boys' High School and a founding father of the University of Tasmania, so it is no surprise that she and several of his other children followed him into careers in education. As a young child Lily grew up in Hobart, her mother was 31 and her father was 57 when Lily and her twin Violet were born. Lily's father was a part-time rector at Holy Trinity Anglican Church, Hobart. Lily grew up in this resolutely low church environment. Life must have been difficult at times for Lily growing up, her father, who had arrived in Tasmania in 1856, was "melancholy in outlook and prone to depression, he had much sadness in his family life.

He mourned the separation from the three daughters left in England and the early death of his son Richard from severe burns. His second daughter Charlotte Maria became of unsound mind, was committed to an institution in February 1872 and died a few years later." Furthermore, "he was charged with assaulting boys with a cane in March 1860 and June 1868, the first case being dismissed and the second settled out of court, but he maintained the school's pre-eminent position in the colony until 1878 when he lost his midlands boarders to Horton College and Launceston Church Grammar School. Thereafter his health declined and in 1885, suffering acute physical pain and mental depression, he surrendered to Christ College, with the shareholders' agreement, all leasehold rights in return for an annuity of £300; the school was closed on 15 August 1885."A "a bright, inquisitive and active child", Lily was schooled by her father and received a Level II mark prize in December 1882. Lily was allowed to sit the major exams as a "trial of strength" in 1884 though she was not eligible for a scholarship.

She came second. She played the violin at school, she would go on playing this instrument, the piano, all of her life, giving occasional public performances at Peppermint Bay and Hobart. For instance, she gave a recital at a church choir fundraising event at her home parish of All Saints in South Hobart less than a year before she died; when her father retired in 1885, he purchased a hotel at Peppermint Bay and converted it into a house which he named "The Cliffs". Lily was to spend her young adulthood here. However, the first indication of Lily's strength of character comes from November 1885, when she was twelve years old. One day, "Miss May Harris, with her two little sisters and Lily, Miss Gaynor, a guest, went down to the beach to bathe, a little while afterwards Mrs. Harris followed them down to look after them. On her way to the beach, when, a little way only from it, her attention was caught by some brushwood and dry grass which she thought might harbour snakes, she accordingly set fire to it with the hope of removing it, was still engaged in the operation, when she – the fire having spread without her noticing it – found that her dress had caught ablaze, that the sleeves were burning.

This was the first intimation she had of her danger, she at once screamed out, rolling herself on the ground, tried to put out the fire in that way. Lily, the younger of the twins, was the only one near enough to assist her mother, rushing up, the little child had the presence of mind to pull off her wet bathing dress and wrap it round her mother's body, thus saving her from much worse injuries than those she received. Mrs Harris was taken home, was found to be suffering from severe burns on the arms and back." Lily's older brother, Henry Vere Poulett-Harris was a gifted footballer and cricketer, he represented both Tasmania and Western Australia in first-class cricket in a career spanning the 1883–84 to 1898–1899 seasons. One early news report described him as a "sterling cricketer and footballer" whilst another described him as a "sterling batsman and good field."Indeed, his obituary states that he was "one of the outstanding athletes in the State, winning great success as a runner and footballer.

He played cricket for the Wellington Club and was regarded as one of the most graceful batsmen in the State. He was a member of the State team when a youth, toured New Zealand with the Tasmanian team under the captaincy of the late Sir George Davies, he met with success as a batsman on the mainland. He was a champion footballer and a member of the Cricketers' Football Club, some of his contemporaries being Messrs. W. H. Cundy, L. H. Macleod, K. E. Burn, A. Stuart and G. Watt; as a runner he defeated many of the recognised champions of his day." It was his interest in sport. Another influence would have been her father who, in 1882, was elected a trustee of the Southern Tasmanian Cricket Association. Furthermore, he encouraged the boys at the high school to compete at sports; as her own obituary notice states, Lily "was a great admirer of athletic exercises believing that it was necessary to develop the ph

Lambert Field (Purdue University)

Lambert Field was a baseball stadium in West Lafayette, Indiana. It was the home field of the Purdue Boilermakers baseball from 1965 until 2012 and held 1,100 people, it was named after former Purdue baseball coach Ward Lambert. Opened in 1965, Lambert Field succeeded the Old Lambert Field as the home of Purdue baseball; the construction of Mackey Arena on the location of Old Lambert Field necessitated the move. In the 1990s, the university undertook three renovation projects on Lambert Field. In 1990, a new press box was added, the seating areas were improved. In 1994, an electronic scoreboard was installed. In 1999, the university adopted a yearly renovation plan for Lambert; the annual improvements have added new fences in the outfield and foul territory, a new backstop, new bullpens, a screen to protect the parking lot beyond the left field fence, an irrigation system. In 2002, the facility hosted the Indiana North-South All-Star Game. In 2012, Purdue won its first Big 10 Regular Season championship since 1909 in the field's final season.

The field was named for former Purdue baseball coach Ward Lambert. During three stints coaching the baseball program, Lambert compiled a 163–156–7 record. For his success in coaching Purdue's basketball program, Lambert was inducted into the Naismith Basketball Hall of Fame in 1960. Purdue built a new baseball venue, Alexander Field, which it began using in 2013. Although the field was planned to open for the 2012 season, construction delays led Purdue to use Lambert Field through the end of the 2012 season. At the end of the season, Lambert was razed to allow space for additional parking for the Student Fitness and Wellness Center