How come Washington is the only city in the District of Columbia ?

There are dozens of cities in DC. Anacostia, Capitol Hill, Adams Morgan, etc.

Washington, D.C., D.C. in full District of Columbia, city and capital of the United States of America. It is coextensive with the District of Columbia (the city is often referred to as simply D.C.) and is located on the northern shore of the Potomac River at the river’s navigation head—that is, the transshipment point between waterway and land transport. The state of Maryland borders the District of Columbia to the north, east, and west, and the state of Virginia borders the District on the southern shore of the Potomac River.

The whole point to DC is to provide somewhere outside the US states for the Federal Government, so no hosting state would have undue leverage or responsibility.

Perhaps a better idea would have been to just carve out 1,000 acres with no residential zoning, just for the federal government to do business.

There used to be.  Georgetown was a separate city, but they merged.  Arlington was part of DC, but is now part of Virginia.

It was established by an act of Congress specifically to be a Federal District and the capital of the Unitrd States and not a part of one of the states. 

As an adjunct to I Like Stories' BEST answer, l wish to add the District is NOT a state because if it were, not being a Federal District, state laws could apply and conflict with operating the country.

FYI- Russia has 4 Federal Cities in federal districts of similar status.

It only has 61 sq miles of land.  

At any rate, it used to be two cities, but Georgetown was merged with Washington in the late 1800s.  Before the land was returned to VA, Alexandria was a city in DC.

With that little land and the specific purpose of being the national capital and not part of a state, it doesn't make sense for it to have multiple cities.  In fact, technically there are no cities in DC.  In 1871, the cities of Washington and Georgetown and the County of Washington were abolished and a single entity was established.

Sigh......

Washington DC was intentionally created to be the capital of the United States of America.  Washington DC has no other purpose than to be the capital.  It was intentionally designated as such to prevent favoritism toward a host state.  Originally the US capital was in Philadelphia.  The District of Columbia is NOT a state or county.  Seriously, you need to go study American history again, this time pay attention.  You should have learned this in 7th grade.