The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker – Campbell – Hausdorff formula: there exists a neighborhood U of the zero element of, such that for u, v in U we have

The addition operation is continuous at 0 if and only if for every neighborhood U of 0 there exists another neighborhood V of 0 such that the Minkowski addition | Minkowski sum V + V is contained in U.

A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous.

# There exists a neighborhood of on which is bounded.

Following its reconstruction after the 1906 Earthquake, the proximity of the nearby docks and fishing wharves attracted a large amount of Italian immigrants that would create the Italian character of the neighborhood that exists to this day.

* There exists a neighborhood of A, called the basin of attraction for A and denoted B ( A ), which consists of all points b that " enter A in the limit t → ∞".

such that for every x ∈ X, there exists an open neighborhood U of x, such that p < sup >− 1 </ sup >( U ) ( the inverse image of U under p ) is a disjoint union of open sets in C, each of which is mapped homeomorphically onto U by p.

* Every cover is a local homeomorphism — that is, for every, there exists a neighborhood of c and a neighborhood of such that the restriction of p to U yields a homeomorphism from U to V. This implies that C and X share all local properties.

For functions of more than one variable, the theorem states that if the total derivative of a continuously differentiable function F defined from an open set U of R < sup > n </ sup > into R < sup > n </ sup > is invertible at a point p ( i. e., the Jacobian determinant of F at p is non-zero ), then F is an invertible function near p. That is, an inverse function to F exists in some neighborhood of F ( p ).

By the theorem, for every point p in R < sup > 2 </ sup >, there exists a neighborhood about p over which F is invertible.

is a linear isomorphism at a point p in M then there exists an open neighborhood U of p such that

In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U.

* We say that x and y can be separated by closed neighborhoods if there exists a closed neighborhood U of x and a closed neighborhood V of y such that U and V are disjoint ( U ∩ V = ∅).

In topology, a branch of mathematics, a point x of a set S is called an isolated point of S if there exists a neighborhood of x not containing other points of S.

Equivalently, z < sub > 0 </ sub > is a ramification point if there exists a holomorphic function φ defined in a neighborhood of z < sub > 0 </ sub > such that ƒ ( z ) = φ ( z )( z − z < sub > 0 </ sub >)< sup > k </ sup > for some positive integer k > 1.

An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π ( P ) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and also denoted by e < sub > P </ sub >.

In other words, there exists a faithfully flat map X → S such that any point in X has a quasi-compact open neighborhood U whose image is an open affine subscheme of S, such that base change to U yields a finite product of copies of GL < sub > 1, U </ sub > = G < sub > m </ sub >/ U. One particularly important case is when S is the spectrum of a field K, making a torus over S an algebraic group whose extension to some finite separable extension L is a finite product of copies of G < sub > m </ sub >/ L.

* for each i and for each point x of X < sub > i </ sub > − X < sub > i − 1 </ sub >, there exists a neighborhood of x in X, a compact ( n − i − 1 )- dimensional stratified space L, and a filtration-preserving homeomorphism.

In the 1840s, the Harlem Railroad made the first rail connection between Manhattan and what became the Woodlawn neighborhood, a connection that still exists via the Woodlawn station on what is now Metro-North Railroad's Harlem Line.

For example, in the US a location can be identified by a zipcode that exists within a neighborhood, city, state, region.

Ghetonia ( Griko: Γειτονία, neighborhood ) is a cultural group based in Calimera, Grecìa Salentina in southern Italy, which exists to preserve the music, poetry, language and folklore of the Griko-speaking people of Salento by documenting the various aspects of the Grecìa Salentina traditional life, history, language and folklore, and has published extensively on these subjects.

In February 2007, a very large, deep circular hole with vertical walls opened in a poor neighborhood in northeast Guatemala city, killing three people.

The execution was carried out at Camp Justice, an Iraqi army base in Kadhimiya, a neighborhood of northeast Baghdad.

Just northeast of Downtown is The Avenues, a neighborhood outside of the regular grid system on much smaller blocks.

North Beach is a neighborhood in the northeast of San Francisco adjacent to Chinatown, Fisherman's Wharf and Russian Hill.

A neighborhood comprising areas within the city limits of northeast El Cajon, as well as areas of unincorporated San Diego County bears the same name.

* Covington Lakes, an upper-middle-class neighborhood of about 500 single-family and townhomes in the northern part of Huntley, northeast of the intersection of Route 47 and Reed Road.

* Huntley Meadows, is a younger middle-class neighborhood of about 200 homes on Huntley's northeast side, south of Algonquin Rd.

* Douglas Avenue, a neighborhood just northeast of town with a mix of housing styles, including ranches, two-stories, townhomes, and a few apartments.

Also farther north in the northeast quadrant is the Kenwood Park which was independent until it was incorporated into the Cedar Rapids city limits and Noelridge Park neighborhood.

Near the northeast corner of town, Route 8A grazes the border ( at which point it is combined with Route 116 ), with a small neighborhood lying around Windsor Pond.

The southern neighborhood of Gilbertville is the location of the intersection of Route 32 and the southern terminus of Route 32A, with Route 32 coming from Ware heading northeast towards Barre, and Route 32A heading due north through the town's center towards the center of Petersham, where its northern end lies.

* Stoney Creek is a neighborhood on the northeast end of the city on the border with Rochester Hills.

Several streets in a residential neighborhood northeast of downtown are named for performers on ABC-TV's Ozark Jubilee, including Nixa native Slim Wilson.

The Montrose neighborhood with its large Victorian houses is in the northeast quadrant.

The Rossville neighborhood and Randolph cemetery are located on the northeast side of the river off North County Road 25-A.

It is in Washington County, to the west of Portland's West Hills neighborhood, to the northeast of Raleigh Hills and south of U. S. Route 26.

It directly borders the Antioch neighborhood of Nashville on the northwest, Smyrna on the southeast, and Percy Priest Lake on the northeast.

In northeast Spokane, the Hillyard neighborhood came about due to the Great Northern Railway yard.

* Sun Valley, a neighborhood in northeast El Paso, Texas

To the north of Chelsea is the neighborhood of Hell's Kitchen, also known as " Clinton ," to the northeast is the Garment District, to the east are NoMad and the Flatiron District, to the southwest is the Meatpacking District and to the southeast is the West Village.

* Natick, a neighborhood in the northeast section of the town.

A news article on August 23, 1901, talked about some of the new construction in the neighborhood in the northeast corner of Tipperary Hill near St. Patrick's Church surrounding Lowell Avenue.

In recent years, long time neighborhood residents of Irish ancestry and local business owners gathered resources and encouraged the city to demolish an old run-down commercial building that was located on the northeast corner of the intersection.

Several parks are located throughout the township: Harrison Hills Park is a county park located in the northeast area of the township ; the Natrona neighborhood has a community playground and riverfront park.