The southwestern border of Connecticut , where it abuts New York State , is marked by a panhandle in Fairfield County , containing Greenwich , Stamford , Fairfield , Westport , Wilton and Darien , housing some of the wealthiest residents in the world . The southwestern border of Connecticut , where it abuts New York State , is marked by a panhandle in Fairfield County , containing the towns of Greenwich , Stamford , New Canaan , Darien and part of Norwalk .