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Definition of domain : 1. Dominion; empire; authority.
2. Landed property; estate; especially, the land about the mansion house of a lord, and in his immediate occupancy; demesne.
3. Ownership of land; an estate or patrimony which one has in his own right; absolute proprietorship; paramount or sovereign ownership.
4. The territory over which dominion or authority is exerted; the possessions of a sovereign or commonwealth, or the like. Also used figuratively.

Synonyms:

field of operations, humankind, sphere of influence, cosmos, heavens, electron orbit, celestial sphere, eye socket, field of force, populace, world, line of business, solid ground, theatre of operations, expanse, farming, subject field, terra firma, field of study, athletic field, specialty, champaign, globe, knowledge domain, scope, human race, earth, commonwealth, scene of action, body politic, humanity, orbital cavity, sphere, dry land, force field, vault of heaven, orbit, department, mankind, macrocosm, terrain, subject, compass, scene, knowledge base, field of view, stadium, surface area, study, welkin, soil, reality, acres, public, battleground, cranial orbit, range, worldly concern, human beings, subject area, playing area, landed estate, reach, airfield, field of operation, playing field, theater of operations, universe, field of honor, plain, landing field, bowl, res publica, creation, flying field, humans, bag, sports stadium, man, empyrean, ambit, demesne, existence, earthly concern, theatre, discipline, domain of a function, field of battle, celestial orbit, firmament

hectare, area, territory, region, sector, rood, district, section, patch, acre, zone

holding, ranch, possessions, land, assets, principal, manor, inheritance, farm, estate, stock, realty, property, wealth, chattel, fixtures, legacy, goods, real estate, title, resources, effects, belongings, possession

colony, commonweal, parcel, duchy, division, earldom, canton, kingdom, locality, quarter, tract, lot, county, plot, municipality, dukedom, state, country, nation, place, riding, Shire, town, neighborhood, borough, empire, city

bailiwick, circuit, circle, ground, battle-ground, arena, dominion, turf, gymnasium, field, hippodrome, stage, campus, enclosure, battlefield, circus, realm, gallery, province, beat, pen, amphitheater, hall, precinct, room, theater, setting, auditorium, locale, ring, Colosseum, courtyard

Usage examples:

• One was thoroughly wearied of the subject before he had the good taste to depart to his own special domain in the room adjoining the parlour. - "Fibble, D. D.", Irvin Shrewsbury Cobb.
• Evelyn, the meadow's mine; I have a right to fence there- between my domain and yours! - "The Celestial Omnibus and other Stories", E. M. Forster.
• It had, no doubt, been abandoned ever since the Navarreins ceased to care for this domain. - "The Village Rector", Honore de Balzac.