What was the name of the colony set up in 1535 in the area know today as mexico?

The colony set up in 1535 in the area know today as Mexico was New Spain.

Explanation:

New Spain was a viceroyalty of Spain during the colonial period in North and Central America, where the today the states of Arizona, California, Colorado, Nevada, New Mexico, and Utah in the United States, all of Mexico, and most of Central America to Costa Rica. It existed from 1535 to 1821, and its capital was Mexico City.

New Spain managed not only land between these boundaries but also the Philippine archipelago in Asia. After the defeat of the Spanish army by the troops of Agustin de Iturbide and Vicente Guerrero, the whole territory (except the Asian archipelago) became part of the Mexican Empire from September 28, 1821.