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Western world has meant various things at various times. In the earliest history of europe it referred to Ancient Greece and the Aegean. During the Roman Empire, it meant the Western Roman Empire (covering the area from Croatia to Britain). At other times, it has meant Western Europe or Europe or Christendom. During the Cold War, it sometimes meant the democratic countries or those allied with the various NATO powers.
In this day and age, it often refers to the places that have a European heritage culture. This european heritage itself is a combination of, Judeo-Christian ethic, Classical Greco-Roman thinking and the cultural practices of the "barbarian" peoples of Northern Europe. Under the strictest definition, it would include North America, Australia and Western/Central Europe: with the identities of Latin America, Russia, Eastern Europe and South Africa being disputed