In mathematics, a Baire space is a topological space such that every intersection of a countable collection of open dense sets in the space is dense. Complete metric spaces and locally compact Hausdorff spaces are examples of Baire spaces according to the Baire category theorem; the spaces are named in honor of René-Louis Baire. In an arbitrary topological space, the class of closed sets with empty interior consists of the boundaries of dense open sets; these sets are, in a certain sense, "negligible". Some examples are finite sets in ℝ, smooth curves in the plane, proper affine subspaces in a Euclidean space. If a topological space is a Baire space it is "large", meaning that it is not a countable union of negligible subsets. For example, the three-dimensional Euclidean space is not a countable union of its affine planes; the precise definition of a Baire space has undergone slight changes throughout history due to prevailing needs and viewpoints. First, we give the usual modern definition, we give a historical definition, closer to the definition given by Baire.
A Baire space is a topological space in which the union of every countable collection of closed sets with empty interior has empty interior. This definition is equivalent to each of the following conditions: Every intersection of countably many dense open sets is dense; the interior of every union of countably many closed nowhere dense sets is empty. Whenever the union of countably many closed subsets of X has an interior point one of the closed subsets must have an interior point. In his original definition, Baire defined a notion of category. A subset of a topological space X is called nowhere dense in X if the interior of its closure is empty of first category or meagre in X if it is a union of countably many nowhere dense subsets of second category or nonmeagre in X if it is not of first category in XThe definition for a Baire space can be stated as follows: a topological space X is a Baire space if every non-empty open set is of second category in X; this definition is equivalent to the modern definition.
A subset A of X is comeagre. A topological space X is a Baire space if and only; the space R of real numbers with the usual topology, is a Baire space, so is of second category in itself. The rational numbers are of first category and the irrational numbers are of second category in R; the Cantor set is a Baire space, so is of second category in itself, but it is of first category in the interval with the usual topology. Here is an example of a set of second category in R with Lebesgue measure 0. ⋂ m = 1 ∞ ⋃ n = 1 ∞ where n = 1 ∞ is a sequence. Note that the space of rational numbers with the usual topology inherited from the reals is not a Baire space, since it is the union of countably many closed sets without interior, the singletons; the Baire category theorem gives sufficient conditions for a topological space to be a Baire space. It is an important tool in functional analysis; every complete metric space is a Baire space. More every topological space, homeomorphic to an open subset of a complete pseudometric space is a Baire space.
In particular, every metrizable space is a Baire space. Every locally compact Hausdorff space is a Baire space. BCT1 shows that each of the following is a Baire space: The space R of real numbers The space of irrational numbers, homeomorphic to the Baire space ωω of set theory The Cantor set Indeed, every Polish spaceBCT2 shows that every manifold is a Baire space if it is not paracompact, hence not metrizable. For example, the long line is of second category; every non-empty Baire space is of second category in itself, every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval. Every open subspace of a Baire space is a Baire space. Given a family of continuous functions fn:X→Y with pointwise limit f:X→Y. If X is a Baire space the points where f is not continuous is a meagre set in X and the set of points where f is continuous is dense in X. A special case of this is the uniform boundedness principle.
A closed subset of a Baire space is not Baire. The product of two Baire spaces is not Baire. However, there exist sufficient conditions that will guarantee that a product of arbitrarily many Baire spaces is again
Poplar Lawn Historic District is a national historic district located at Petersburg, Virginia. The district includes 372 contributing buildings and consists of mid- to late-19th-century, single-family residences grouped about a two-block green, it includes notable examples of Greek Revival, Colonial Revival, Second Empire, Italianate style residential architecture. Notable buildings include the Bolling-Zimmer House, St. Stephen's Church, Zion Baptist Church, William T. Double House, the Waterworks, Dr. Robert Broadnax House, Market Street Methodist Church Parsonage, Maurice Finn House, the Frank M. D'Alton Double House; the district includes Poplar Lawn Park, featuring a stone basin of uncertain age, five feet across, having an oval-shaped depression a foot wide and a foot deep. It is traditionally known as "Pocahontas' bath", though there is no proof she used it, it was listed on the National Register of Historic Places in 1980, with a boundary increase in 2006
Todd Shelton is an American fashion brand founded in 2002 by Tennessee native Todd Shelton. The brand's collections are manufactured within the company's New Jersey headquarters and sold online; the Todd Shelton factory and showroom are headquartered New Jersey. Todd Shelton designs and manufactures made-to-order jeans, pants and t-shirts for men; the company have a U. S. factory that offers e-commerce customers ways to personalize their button-down shirts or jeans with multiple-fit choices, such as straight or tapered leg, or the length of a sleeve, or shirttail. Todd Shelton is an American-Made Men’s Clothing Brand; the brand's manufacturing puts a twist on online shopping. At the intersection of off-the-rack and bespoke tailoring. Todd Shelton’s company owns 55 sewing machines, a fusing machine and a handful of other specialty machines that help make their line of T-shirts, button-downs and jeans; the brand began by outsourcing, but in 2006, Todd Shelton moved all manufacturing to factories in the USA.
As Shelton grew frustrated by the supply chain and being dependent on manufacturer’s timelines and frequent fickleness, he scaled things back and, in 2012, opened his own factory. Official website
"Death Is the Only Answer" is a special mini-episode of the British science fiction television series Doctor Who, first broadcast on BBC Three on 1 October 2011. It was written via a "Script to Screen" competition in which junior schools were asked to write a script including the Eleventh Doctor and an enemy of his; the competition was won by the children of Oakley CE Junior School. In the mini-episode, the Doctor's fez calls its former owner Albert Einstein into the TARDIS. Einstein carries a mysterious liquid; the episode opens with the Doctor in the TARDIS reunited with his fez from "The Big Bang". While reminiscing about it he trips over, knocking the fez off his head and hitting a lever on the console; the fez vanishes and Albert Einstein appears with the fez on his head. Einstein greets the Doctor and explains that he had been working on a time machine and thought he'd found the vital part of it — a mysterious liquid which Einstein believes to be "bionic fusion liquid"; the Doctor offers to run some tests on it but Einstein insists upon testing it himself and turns to walk away from the console.
However, the liquid changes Einstein into an Ood. For several seconds the "Albert-Ood" repeats confusing the Doctor. A white cylinder vortex appears and the "Albert-Ood" walks towards it and somehow manages to turn back into Einstein. Einstein asks to go home so the Doctor drops him off in 1945; the Doctor leaves for another adventure, while a bit of the liquid is left on the floor, moving slowly. "Death Is the Only Answer" is the winner of BBC Learning's "Script to Screen" writing competition, which began on 28 April 2011 and encouraged children aged 9 to 11 to collaborate on a three-minute script for the Eleventh Doctor. The script could feature a new human character and an enemy of an Ood, Cyberman, or Weeping Angel. BBC Learning's website contained learning resources for teachers and pupils to exercise their writing skills, while Matt Smith, Karen Gillan and Arthur Darvill presented a series of videos in character as the Doctor, Amy Pond and Rory Williams respectively. Videos were produced with advice about how to write a good script and incorporating character and stage directions.
Response to the competition was great, with 292,000 downloads of the online teaching material. The children's teacher, Kevin Downing, had told all the upper school children about the competition at an assembly, asking them to come up with a good plot in small groups, he said. Being Doctor Who fans, they had great fun deliberating over how the Doctor would react to things, what he would say and imagining how it would look on screen. None of us imagined the school would win the competition of course, but it was exciting to have entered and have the finished script to look back on in years to come."The chosen script was written by four pupils from Oakley CE Junior School: Adam Shephard, Ben Weston, Daniel Heaton, Katie Hossick. They chose an Ood to feature in the mini-episode after deciding that the Cybermen "walk a bit too slowly", the Judoon are "not scary", a Weeping Angel "wouldn't do anything"; the appearance of Albert Einstein came about. The winners were selected by showrunner Steven Moffat, BBC Learning controller Saul Nassé, Doctor Who executive producers Piers Wenger and Beth Willis.
Of the judging, Moffat said, I loved the shortlisted scripts, there was so much skill and enthusiasm on display that it was genuinely very difficult to judge. There was some really skilled writing, it was exiting how they caught the voice of the Doctor and how they used the always stringent limitations of Doctor Who to their advantage exciting. Matt Smith was impressed with the chosen script, Albert Einstein's role in the mini-episode. Smith, who had written his own stories about Einstein meeting the Doctor, said that the story was "brilliantly realised, they did it with aplomb."The winners were taken to the Upper Boat Studios complex near Cardiff in Wales to see their episode being filmed, while the Doctor Who Confidential team was there to document the transition from script to screen. Nickolas Grace appeared in the Seventh Doctor audio drama Bang-Bang-a-Boom! where he played Loozly, as the Time Lord Straxus in the Eighth Doctor audio dramas Human Resources, Sisters of the Flame and Vengeance of Morbius.
Hatfield is an unincorporated census-designated place, in the town of Komensky, Jackson County, United States. As of the 2010 census, its population is 141. Hatfield has an area of all of it land, it is located on the shores of an impoundment of the Black River. The dam forming the impoundment releases water back into the river channel and a diversion channel for a hydroelectric powerhouse. Hatfield is a tourist community that claims a population of 50 in the winter. Two county campgrounds are located in the community, the Levis/Trow trail system is two miles to the north. Hatfield was founded by Norbert St. Germaine in 1836; the city was supported in the 19th century by the logging and lumber industry, as the Black River was a primary avenue for delivery of logs from central Wisconsin to the Mississippi River valley. The Green Bay and Western Railroad arrived in 1872. Hatfield was the birthplace of Mitchell Red Cloud, Jr. of the Ho Chunk Nation, awarded the Medal of Honor posthumously for his actions in the Korean War.