In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure, used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "x is closer to a than y is to b". By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A, or that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone. There are three equivalent definitions for a uniform space, they all consist of a space equipped with a uniform structure.

This definition generalizes the presentation of a topological space in terms of neighborhood systems. A nonempty collection Φ of subsets U ⊆ X × X is a uniform structure if it satisfies the following axioms: If U ∈ Φ Δ ⊆ U, where Δ = is the diagonal on X × X. If U ∈ Φ and U ⊆ V ⊆ X × X V ∈ Φ. If U ∈ Φ and V ∈ Φ U ∩ V ∈ Φ. If U ∈ Φ there is V ∈ Φ such that V ∘ V ⊆ U, where V ∘ V denotes the composite of V with itself. If U ∈ Φ U − 1 ∈ Φ, where U − 1 = is the inverse of U; the non-emptiness implied by property taken together with and state that Φ is a filter on X × X. If the last property is omitted we call the space quasiuniform; the elements U of Φ are called entourages from the French word for surroundings. One writes U =. On a graph, a typical entourage is drawn as a blob surrounding the "y = x" diagonal. If ∈ U, one says that y are U-close. If all pairs of points in a subset A of X are U-close, A is called U-small. An entourage U is symmetric if ∈ U when ∈ U; the first axiom states that each point is U-close to itself for each entourage U.

The third axiom guarantees that being "both U-close and V-close" is a closeness relation in the uniformity. The fourth axiom states that for each entourage U there is an entourage V, "not more than half as large"; the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in x and y. A fundamental system of entourages of a uniformity Φ is any set B of entourages of Φ such that every entourage of Ф contains a set belonging to B. Thus, by property 2 above, a fundamental systems of entourages B is enough to specify the uniformity Φ unambiguously: Φ is the set of subsets of X × X that contain a set of B; every uniform space has a fundamental system of entourages consisting of symmetric entourages. Intuition about uniformities is provided by the example of metric spaces: if is a metric space, the sets U a = where a > 0 form a fundamental system of entourages for the s

Dundee Beach is a locality in the Northern Territory of Australia, located 59 km southwest of the territory capital of Darwin. The 2016 Australian census, conducted in August 2016 reports that Dundee Beach had 207 people living within its boundaries. Dundee Beach consists of land bounded by the coast of Fog Bay in the west and a subsidiary inlet of Bynoe Harbour in the east and the waters to the west of the coastline for a distance of 3 nautical miles. Dundee Beach and the other two nearby localities prefixed with the name "Dundee" are named after the sub-divisions with these names; the name "Dundee" is believed to be derived from Crocodile Dundee. The boundaries for Dundee Beach were gazetted on 29 October 1997 with the gazettal being revoked on 3 April 2007 with new boundaries being gazetted on 4 April 2007. Development in the area began in the 1990s as a weekend retreat for Darwin residents; the local economy is still dependent on tourism, with fishing being the main attraction. Community services and amenities include a small school established in 1998, a volunteer fire brigade and several sporting and social clubs.

One of Australia's earliest artifacts of European origin was found by a 13-year-old school boy on Dundee Beach in 2010. Believed to be a Portuguese Swivel gun dating from the 1500s, the discovery prompted international media attention and renewed speculation about early Portuguese exploration of Australia prior to the arrival of Captain Cook on the East Coast in 1770; however this claim has been disputed by the Museum and Art Gallery of the Northern Territory following analysis of sand inside the gun's barrel dating it closer to 1750. The museum instead speculated it was dropped overboard by Makassan traders who were known to visit the area to harvest Sea cucumber and trade with local Aboriginal peoples; this indicates the gun was more a copy of a Dutch or Portuguese design, manufactured in South East Asia. Dundee Beach is located within the federal division of Lingiari, the territory electoral division of Daly and the unincorporated areas of the Northern Territory; the possibility of Dundee Beach and adjoining localities being included in an expanded Coomalie Shire was discussed in 2009 without any decisive outcome.

However, a poll of residents carried out in 2012 showed that a majority were in favour of retaining the unincorporated area status. The Dundee Progress Association, a volunteer association, represents the interests of residents in the locality and those in Bynoe Harbour, Dundee Downs and Dundee Forest to the organizations including the Northern Territory Government

The Indian pariah dog known as the South Asian pye dog and Desi Dog, is a landrace of dog native to the Indian subcontinent. They have erect ears, a wedge-shaped head, a curved tail, it is trainable and used as a guard dog and police dog. This dog is an example of an ancient group of dogs known as pariah dogs, it is possible. Though most street dogs in the Indian subcontinent are in fact Indian pye-dogs, the names for this breed are erroneously used to refer to all urban South Asian stray dogs despite the fact that some free-ranging dogs in the Indian subcontinent do not match the "pariah type" and may not be pure indigenous dogs but mixed breeds around locations where European colonists settled in India, due to admixtures with European dog breeds, it is a medium-sized dog of square to rectangular build and short coat. The dog has a double coat, a coarse upper coat, a soft undercoat; the most observed colours are browns, ranging from dark to reddish-brown, with or without white markings. Solid blacks are rare.

Shaded coats, solid white and dalmatian-type spotting are never seen in pure populations. These may be a sign of mixing with modern breeds, as they are only seen in dogs in cities and other sites where non-native dogs have been introduced; the head is wedge-shaped. The muzzle is pointed and is of equal or greater length than the head; the neck is noble and the forequarters are erect. Hindquarters are minimally angled; the trot is short. The eyes are dark brown in colour; the ears are held erect and are pointed at the tips, with a broad base, set low on the head, the tail is curled and held high when excited. The namesake of this breed was given during the British Raj in India after the Pariah tribe of the Madras Presidency. From the Anglo-Indian word pye or paë and Hindi pāhī meaning'outsider', the Indian pariah dog is sometimes referred to as the pye-dog and the Indian native dog, it is popularly known as Desi Dog, as well as the Indi-dog or In-dog. People in Northern India and Pakistan call it "desi kutta".

In Bengali they are named as "Neri Kukur". In Himachali, they are named as "luru". In Assamese language these native dogs are termed as bhotua kukur; this breed is known as "naadan" in Malayalam, "naatu naai" in Tamil, "oora kukka" in Telugu language in Andhra Pradesh and Telangana, "ooru naai" in Kannada and "Bhusya Kukkur" in Nepal. It was referred to in the works of Rudyard Kipling as the "yellow pariah dog"; the pariah dog of India is an ancient autochthonous landrace, found all over India, Bangladesh and beyond South Asia. A Indian pariah dog skull was discovered in the ancient Indian site of Mohenjo-daro and prehistoric rock art depicting this breed has been found in the Bhimbetka rock shelters, it was featured on National Geographic Channel's film, Search for the First Dog along with the other related ancient types such as the Canaan Dog of Israel and the Australian dingo. The Indian pye-dog was introduced to the Andaman Islands with the establishment of a penal colony there, dogs having been unknown to the native Andamanese.

Despite the Indian pariah dog being intelligent and trainable, the breed was intentionally downplayed during the British Raj by merchants who wished to sell their foreign breeds within the country. Their popularity in the West in recent years, has resulted in hundreds of dogs being exported out of the Indian subcontinent. In 2015, a breed standard was published in the Indian Kennel Gazette, the publication of Kennel Club of India, the dog has been recognized by the Primitive and Aboriginal Dog Society, a worldwide grouping of enthusiasts based in the US; some in the society view these dogs as a risk citing their increasing population in India in recent years. They consider these dogs as nuisance owing to constant barking and biting people. Most of these attacks are caused due to human provocation like hitting them with sticks or throwing stones at them; the numbers of dog bites and deaths due to dog attacks are increasing every year. Since these dogs are not vaccinated, they carry rabies. Pariah dogs are alert and social.

They are used as guard dogs and police dogs, being territorial and defensive. They need good socializing as pups and do well with families and children if provided with such socialization, they are intelligent and trainable. Being a evolved breed, they have few health concerns and thrive with minimal maintenance in suitable climates; the skin needs little grooming and the dogs themselves are clean. They have little body odour. Genetic health ailments like hip dysplasia are rare, since there is no inbreeding and the dominant genes that aid their survival are selected over time. Most of their deaths occur due to accidents on the roads and railway tracks, not getting food or drinking polluted water, tumors in the body, getting bit by snakes or being beaten up by humans. New Guinea singing dog Dingo Basenji Telomian Sinhala Hound INDog: The Indian Pariah Dog, at InDog.co.in Primitive & Aboriginal Dog Society Life as a pariah sees it - Times of India