What country had settlements established in the southwest united states in the 1500s?

The country that had settlements established in the Southwest United States in the 1500s was Spain.

Explanation:

The territory that currently forms part of the American Southwest (the states of Arizona, California, Colorado, Nevada, New Mexico, Utah, Oklahoma and Texas) was part of the colony of New Spain, which lasted from 1535 to 1821, when Mexico became independent. Later, these territories were acquired by the United States as a result of the Mexican-American War.

The Battle of Saratoga did not allow the British to cut off New England from the rest of the colonies. 

British general John Burgoyne had proposed the plan to isolate New England from the rest of the colonies.  The plan had about 8,000 British troops invading into New York from Canada and taking control of the Hudson River.  The plan was initially successful -- the British took Fort Ticonderoga in June of 1777.  But General Burgoyne overextended his access to supplies as he came further into Patriot territory, and the colonial forces were able to cut off his supply line.  After a couple of difficult battles, Burgoyne's eventual surrender of his troops came with the loss at Saratoga in October, 1777.