down town vs city

down town and city both are nouns.

Word	Noun	Adjective	Verb	Adverb
down town	Yes	No	No	No
city	Yes	No	No	No

down town (noun)	city (noun)
the center of a city	people living in a large densely populated municipality
	a large and densely populated urban area; may include several independent administrative districts
	an incorporated administrative district established by state charter