Yes, Vatican City is a city. The Vatican is a small area set aside in the city of Rome and it the seat of the Catholic church. It is where the Pope resides and does his business for the people of his church. It is mainly made up of St. Peter's Church, the gardens of the pope, the Vatican museum's, the Sistine Chapel, and the offices related to the governance of the church. It has it's own security and guards. There are other parts of the Vatican beyond the small area where it is located such as Castel Gandolfo.
No. The Vatican is located in Rome, Italy and is the smallest city state in world. It is the home of the Pope and the Catholic Church. Located in the Vatican is St. Peter's that was designed by Michelangelo and the Sistine Chapel that was painted by him. Buried within St. Peter's is the tomb of Peter and past popes. Also there is 4 museums, and vast amounts of art and antiquities. There is also a library filled with books from through the ages and it has it's own banking system. I have been there three times and still want to go back.
No, however there is a synagogue in Rome that is about 2 miles from the border of Vatican City.
No, but there are several a short distance from the Vatican in the city of Rome.
The Vatican is totally surrounded by the city of Rome, Italy.
No, but it is the smallest country. Vatican City is a country, and Vatican City is its capital. Many rural cities are smaller.
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Important cities such as Rome, Assisi and Vatican City.
Apart from Rome itself, Vatican City
Mecca Jerusalem Vatican City
Rome (Italy) and Vatican City (Vatican City) are the two closest capitals of Europe (even of the world). The distance between them is 0 km. However Vatican city is a sovereign state and not a true capital of a country. The two closest capital cities in Europe are Vienna and Bratislava. Vatican City is a sovereign state and its capital is Vatican City. The state "Vatican City" comprises of more territory than just the capital ( see http://en.wikipedia.org/wiki/Properties_of_the_Holy_See ), so the first statement is still valid.
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The Vatican is an independent country with its own government. Size or number of cities are of no importance when classifying a country. A country can be of any size.
Vatican
Vatican City is the capital of Vatican City the country.
There are no major cities in Rome - Rome is a city, the capital city if Italy. (It you meant to ask - "What are the major cities of the ancient Roman Empire" - then ask this as a separate question).