SUMMARY / RELATED TOPICS

Homeomorphism

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, from a topological viewpoint they are the same; the word homeomorphism comes from the Greek words ὅμοιος = similar or same and μορφή = shape, introduced to mathematics by Henri Poincaré in 1895. Speaking, a topological space is a geometric object, the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other. However, this description can be misleading; some continuous deformations are not homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms are not continuous deformations, such as the homeomorphism between a trefoil knot and a circle.

An often-repeated mathematical joke is that topologists can't tell the difference between a coffee cup and a donut, since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in the cup's handle. A function f: X → Y between two topological spaces is a homeomorphism if it has the following properties: f is a bijection, f is continuous, the inverse function f − 1 is continuous. A homeomorphism is sometimes called a bicontinuous function. If such a function exists, X and Y are homeomorphic. A self-homeomorphism is a homeomorphism from a topological space onto itself. "Being homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes; the open interval is homeomorphic to the real numbers R for any a < b.. The unit 2-disc D 2 and the unit square in R2 are homeomorphic. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ↦.

The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is a homeomorphism between the domain of the parametrization and the curve. A chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space; the stereographic projection is a homeomorphism between the unit sphere in R3 with a single point removed and the set of all points in R2. If G is a topological group, its inversion map. For any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, the inner automorphism y ↦ x y x − 1 are homeomorphisms. Rm and Rn are not homeomorphic for m ≠ n; the Euclidean real line is not homeomorphic to the unit circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2 but the real line is not compact. The one-dimensional intervals and are not homeomorphic because no continuous bijection could be made; the third requirement, that f − 1 be continuous, is essential.

Consider for instance the function f: [ 0, 2 π ) → S 1 defined by f = {\textstyle f=(\cos \p

Pennsylvania Route 321

Pennsylvania Route 321 is a 43.8-mile-long state highway located in Elk and McKean counties in the U. S. state of Pennsylvania, maintained by the Pennsylvania Department of Transportation. The southern terminus is at U. S. Route 219 in the community of Wilcox; the northern terminus is at PA 346 within the Allegheny National Forest. PA 321 heads northwest from Wilcox through rural areas to the borough of Kane, where it forms a brief concurrency with US 6. North of here, the route heads through the national forest and runs along the shore of the Allegheny Reservoir. PA 321 runs east with PA 59 before winding north through more forest to its northern terminus. A portion of the route along the Allegheny Reservoir is designated as the Longhouse National Scenic Byway, a Pennsylvania Scenic Byway and National Forest Scenic Byway; the road between Wilcox and Kane was designated as part of Legislative Route 97 in 1911 and as part of PA 6 in 1924. US 119 became concurrent with PA 6 in 1926 before US 219 replaced both designations on this stretch of road two years later.

The road between Kane and Kinzua was built in the late 1920s and became an extension of PA 68 in 1935. In 1952, US 219 was relocated off the road between Kane. Plans were made to construct the Kinzua Dam in 1960, several new roads would need to be built to accommodate the reservoir including a relocation of PA 68. In 1961, the PA 321 designation was approved for the unnumbered road between Wilcox and Kane and PA 68 between Kane and Kinzua in order to provide an access road to the planned recreation area. In the mid-1960s, improvements were planned for PA 321; the PA 321 designation north of Kane was removed in 1966. In the late 1960s, the section between 4 miles north of Kane and PA 346 was constructed, including a new alignment between Red Bridge and PA 59. Work on the 4-mile segment north of Kane took place in the early 1970s. In Kane, PA 321 was relocated from following US 6 through the downtown area to use Hacker Street to the east, with reconstruction finished in 1973; the road between Wilcox and Kane, including a bypass of the former, was rebuilt in the mid-1970s.

In 1974, PA 321 was extended from US 6 in Kane north to PA 346. PA 321 begins at an intersection with US 219 in the community of Wilcox in Jones Township, Elk County, heading north on two-lane undivided Buena Vista Highway; the road bypasses the center of Wilcox to the west. The highway curves northwest into forests; the route runs through Dahoga and continues along the eastern border of Allegheny National Forest, with a Buffalo and Pittsburgh Railroad line a short distance to the west of the road. PA 321 enters Wetmore Township in McKean County and becomes Brickyard Road, departing the Allegheny National Forest; the road continues through rural land, passing through East Kane. The route heads into the borough of Kane and becomes Westerberg Way, running through industrial areas as it crosses an abandoned railroad line and comes to an intersection with US 6. Here, PA 321 turns west for a concurrency with US 6 on Biddle Street turning north-northwest onto Hacker Street, where it is lined with homes.

After passing Glenwood Park, the road curves northeast as Kinzua Avenue and leaves Kane for Wetmore Township, becoming Kane-Marshburg Road and heading north-northwest into wooded areas. The route re-enters the Allegheny National Forest and continues northwest, crossing into Hamilton Township and intersecting Longhouse Drive. PA 321 comes to a bridge over the Kinzua Creek arm of the Allegheny Reservoir near the Red Bridge Recreation Area, which contains a picnic area, a campground, a bank fishing area. From here, the highway follows the northeastern shore of the reservoir, passing through Dunkle Corners; the road turns to the east and runs along the south shore of Chappel Bay, soon heading away from the reservoir and turning northeast. The route continues into Corydon Township and reaches an intersection with PA 59 near the Bradford Ranger Station. At this point, PA 321 turns east for a concurrency with PA 59 and the name remains Kane-Marshburg Road; the road leaves the national forest and heads through the community of Klondike before PA 321 splits to the northwest onto Sugar Run Road in Lafayette Township.

The road runs through more of the Allegheny National Forest. The route passes to the north of a section of the Allegheny Reservoir again and turns to the northeast. PA 321 winds north and serves the Tracy Ridge Trailhead, which consists of a campground and hiking trails leading to the reservoir; the route comes to its northern terminus at PA 346 a short distance south of the New York border. The portion of PA 321 between Longhouse Drive and PA 59 is part of the Longhouse National Scenic Byway, a Pennsylvania Scenic Byway and National Forest Scenic Byway that encircles the Kinzua Creek arm of the Allegheny Reservoir and serves multiple recreational areas. In 2015, PA 321 had an annual average daily traffic count ranging from a high of 4,300 vehicles along the US 6 concurrency in Kane to a low of 100 vehicles between PA 59 and PA 346; the portion of PA 321, concurrent with US 6 is part of the National Highway System. The PA 321 designation was approved by the Pennsylvania Department of Highways in 1961, replacing what was known as "Old 219" between Wilcox and Kane and a portion of PA 68 between Kane and Kinzua.

Plans for improving PA 321 were announced in 1963, including a new alignment north of Red Bridge to PA 59 that would follow the shore of the Allegheny Reservoir. In 1965, construction began on a portion of the road between north of Kan

Flávio Murtosa

Flávio Teixeira, known as Flávio Murtosa or Murtosa, is a Brazilian former professional footballer and professional football manager. Murtosa has a long friendship with FIFA World Cup winning manager Luiz Felipe Scolari and works as his assistant coach. Murtosa began playing professional football as a right winger with his hometown's Esporte Clube Pelotas in 1967, he played in the Campeonato Gaúcho with Pelotas, before signing with Maranhão Atlético Clube in 1975. After one season, he returned to Pelotas where he would play until a knee injury forced him to retire at age 26. Al QadisiyaKuwait Emir Cup: 1989CriciúmaCopa do Brasil: 1991GrêmioCampeonato Brasileiro Série A: 1996 Copa do Brasil: 1994 Campeonato Gaúcho: 1987, 1995, 1996 Copa Libertadores de América: 1995 Recopa Sudamericana: 1996PalmeirasCopa do Brasil: 1998, 2012 Copa Mercosur: 1998 Copa Libertadores: 1999 Torneio Rio-São Paulo: 2000CruzeiroCopa Sul-Minas: 2001BunyodkorUzbek League: 2009 KuwaitGulf Cup of Nations: 1990BrazilFIFA World Cup: 2002 FIFA Confederations Cup: 2013