In mathematics, a topological space is called separable if it contains a countable, dense subset. Like the other axioms of countability, separability is a "limitation on size", not in terms of cardinality but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, in general stronger but equivalent on the class of metrizable spaces. Any topological space, itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the real line, in which the rational numbers form a countable dense subset; the set of all vectors ∈ R n in which r i is rational for all i is a countable dense subset of R n. A simple example of a space, not separable is a discrete space of uncountable cardinality.

Further examples are given below. Any second-countable space is separable: if is a countable base, choosing any x n ∈ U n from the non-empty U n gives a countable dense subset. Conversely, a metrizable space is separable if and only if it is second countable, the case if and only if it is Lindelöf. To further compare these two properties: An arbitrary subspace of a second-countable space is second countable. Any continuous image of a separable space is separable. A product of at most continuum many separable spaces is separable. A countable product of second-countable spaces is second countable, but an uncountable product of second-countable spaces need not be first countable. We can construct an example of a separable topological space, not second countable. Consider any uncountable set X, pick some x 0 ∈ X, define the topology to be the collection of all sets that contain x 0; the closure of x 0 is the whole space, but every set of the form is open. Therefore, the space is separable; the property of separability does not in and of itself give any limitations on the cardinality of a topological space: any set endowed with the trivial topology is separable, as well as second countable, quasi-compact, connected.

The "trouble" with the trivial topology is its poor separation properties: its Kolmogorov quotient is the one-point space. A first-countable, separable Hausdorff space has at most the continuum cardinality c. In such a space, closure is determined by limits of sequences and any convergent sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X. A separable Hausdorff space has cardinality at most 2 c, where c is the cardinality of the continuum. For this closure is characterized in terms of limits of filter bases: if Y ⊆ X and z ∈ X z ∈ Y ¯ if and only if there exists a filter base B consisting of subsets of Y that converges to z; the cardinality of the set S of such filter bases is at most 2 2 | Y |. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection S → X when Y ¯ = X; the same arguments

Mike Lynch is an Irish former rugby union player. Lynch played club rugby for Young Munster in the All-Ireland League, was part of the team that lost finals of the Munster Senior Cup to Shannon in 2001 and 2002, he went on the win 21 caps for Munster between 1999, scoring 53 points for the province. He was called up to the senior Ireland squad for the 1998 Five Nations Championship, despite being on the bench for the game against France he was never capped by Ireland at senior level. After his playing career, Lynch was a PE teacher at St Clement's Redemptorist College in Limerick, he was an assistant coach at Young Munster. Munster Profile Mike Lynch at European Professional Club Rugby

A flat bar road bike called a fitness bike, is a hybrid bike optimized for road usage or a road bike with a flat handlebar in place of a drop bar. Frame construction and geometry borrow from conventional road design; the frame is constructed to a light or middleweight standard with a shape that promotes an aggressive, aerodynamic posture suited to riding at higher speeds. There is no conventional suspension for either wheel, though the front fork may use carbon or steel to quell vibration. Wheel size is universally 700c with a width of 28mm to 32mm, somewhat wider than the 23mm to 25mm road bike standard; the drivetrain of a flat bar bike borrows features from multiple bike styles, pairing the trigger-shifting approach of mountain bikes with the taller cassette ratios of road bikes. The brakes of purpose-built flat-bar designs tend to be linear-pull, a mechanism nonexistent in road bikes and displaced by discs with mountain bikes. Disc brake penetration in road cycling continues to increase, with flat bar and cyclocross bikes leading the curve.

Relative to more upright hybrids, flat bar road bikes are lighter and more efficient to pedal, though less so than a drop bar bike by equal measure. Drop bar bikes will have considerable aerodynamic advantages above about 15 MPH. Conventional hybrids will have a measure of off-road capability lacking in entirety from flat bar bikes. For efficient speed with the familiarity and stability of a flat bar, the flat bar road bike is an optimal compromise. Outline of cycling

Chahta Tamaha was an important town in the Choctaw Nation, Indian Territory that served as the Choctaw capital from 1863 to 1883. The town grew up around the Armstrong Academy; the townsite is located in present-day Bryan Oklahoma. Today nothing is left of the Academy. However, the Armstrong Academy Site is listed on the National Register of Historic Places. Armstrong Academy was founded as a school for Choctaw boys in 1844, it was named after a popular agent of the Choctaws. The site was selected because there was a good fresh water spring with enough current to run a gristmill. A large wood supply was available; the first classroom buildings and dormitories were built of logs from the area. In the late 1850s a brick building replaced the log building. A two-story brick addition was added later. A trading post and church were established early on; as a school the average attendance was about 65 students though in 1859 it had about 100 students."The mission was transferred from the American Indian Mission Association to the Domestic Board of Southern Baptist Convention."

The Baptist Missionary Society of Louisville, Kentucky directed activities until 1855. In that year it was turned over to the Cumberland Presbyterian Board of Foreign and Domestic Missions who directed it until the school closed in 1861 at the outbreak of the Civil War. Allen Wright a Choctaw Presbyterian missionary, served as principal instructor at the academy during 1855–1856. Armstrong Academy was located in Blue County, Choctaw Nation until 1886, when the area became part of a newly formed county, Jackson County. During the Civil War the academy closed. Part of the building was used as a Confederate Hospital; the Choctaw Council met there in 1863, the Choctaw capital was transferred there during the same year. The United Nations of Indian Territory delegates met there with the Confederacy to plan war strategy. Commercial activities increased during that time. Chahta Tamaha remained the capital of the Choctaw Nation until 1883, when the capital was relocated to Tuskahoma. In that same year the Armstrong Academy again became a school.

Admission was limited to orphaned boys. The Armstrong Academy was destroyed by fire in February 1921; the Federal government refused to rebuild it, today the area has reverted to its original state as a deserted pasture. Nothing remains of the town but rubble from the Armstrong Academy. Wright, Muriel H. "Historic Spots in the Vicinity of Tuskahoma". Chronicles of Oklahoma 9:1 27-42. Wright, Muriel H. George H. Shirk, Kenny A. Franks. Mark of Heritage. Oklahoma City: Oklahoma Historical Society, 1976. Armstrong Academy Cemetery

Back on the Streets is the debut album by American rock singer/guitarist Donnie Iris, released in 1980. The single "Ah! Leah!" was a hit for Iris, reaching #29 on the U. S. Billboard Hot 100 chart and #19 on the U. S. Billboard Top Tracks chart. Side one "Ah! Leah!" – 3:46 "I Can't Hear You" – 3:40 "Joking" – 4:04 "Shock Treatment" – 3:48 "Back On The Streets" – 3:36Side two "Agnes" – 3:30 "You're Only Dreaming" – 4:45 "She's So Wild" – 2:35 "Daddy Don't Live Here Anymore" – 3:49 "Too Young to Love" – 5:31Re-released October 1980 as MCA 3272 with sides one and two reversed. Donnie Iris - lead and background vocals Mark Avsec - piano, vocals Marty Lee Hoenes - acoustic and electric guitars Albritton McClain - bass guitar Kevin Valentine - drums Kenny Blake - saxophone Robert Peckman - bass on "Shock Treatment" Executive Producer: Carl Maduri Producer: Mark Avsec Engineer: Jerry Reed Album - Billboard Singles - Billboard

Semi Kunabuli Kunatani is a Fiji rugby union player. He plays for the Fiji sevens team and Premiership Rugby side Harlequin F. C.. Before signing for Harlequin F. C. he played for Stade Toulousain. Kunatani debuted for Fiji in 2013 Dubai Sevens tournament. Kunatani started his career playing rugby in the local 7's competition, he played for the Yamacia 7's side and was selected by Ben Ryan to represent the Fiji National Sevens side in Dubai 2013, following the 2013 win at the Bayleys Fiji Coral Coast Sevens where he was a stand out player. Semi's highs this 2014–2015 World Rugby Sevens season include having scored a total of 37 tries in the series thus far, being named in three World Rugby Dream Teams' and most Semi was recognised by Sir Gordon Tietjens when he was named by Sir Gordon in his Hong Kong 7s Dream Team. Listing Semi Kunatani on his bench in an all time Best Sevens team, joining the likes of Waisale Serevi, Jonah Lomu, David Campese, Eric Rush and many other greats of the game. Semi has signed with Toulouse in the French T14, beginning his career in the 2015–16 season and will continue to represent Fiji Sevens if and when selected by national coach Ben Ryan.

Semi became an Olympian at Rio 2016 playing in all 6 matches of the Olympics and helping Fiji win their first Olympic gold medal when Fiji thrashed Great Britain 43-7 in the final. Semi was a major factor in the final playing a significant role in 4 tries of the first half; this saw Semi join Osea Josua Tuisova in the Rio 7s Dream Team. After the 2016 Summer Olympics, Kunatani was awarded the Officer of the Order of Fiji. Toulouse Profile sporple bio Zimbio bio The X factor Kunatani gets clattered by Lee Jones Semi Kunatani at Olympics at Sports-Reference.com