This country's capital's short name is cdmx. cdmx was built on a lake. what is the name of the country?

The country is Mexico and CDMX means "Ciudad de Mexico".

Explanation:

The country is Mexico and CDMX means "Ciudad de Mexico". CDMX is the short name of the Capital City of Mexico, which used to be called "Distrito Federal" up until the year 2016. In that year its name was definitely changed to CDMX. It's a really big city, the biggest of Latin-America and one of the ten most populated cities in the world. It was built on a lake, called "Lago de Texcoco" around the 14th century.