Final answer:
The region of Australia known for its urban rim, where cities are densely populated, is New South Wales, home to Sydney.
Step-by-step explanation:
The region of Australia with the urban rim is New South Wales. This state is home to the largest city in Australia, Sydney, which is positioned at the heart of the main core area in the eastern region of the country, extending from Brisbane to Adelaide. The urban rim is characterized by high population density and significant urbanization, with a core-periphery spatial relationship in which the core areas like Sydney hold the power, wealth, and influence.