In order to understand this question, let us first think about the meaning of the word “cosmopolitan.” The word has various meanings, all having to do with diversity and with being comfortable with diversity. The definition that is most relevant here is something like “being worldwide rather than regional or provincial.”
When we look at the map of the Persian Empire, we can see why it would be cosmopolitan. Of course, it was not worldwide in our modern sense. However, it was very large. It included many different areas at a time when transportation and communication were not very fast. In those days, there would have been many different cultures within a given area than we have now in our more globalized world. The Persians ruled an area that stretched from Afghanistan to Turkey to Egypt. Even today, this would be a large country that would be very cosmopolitan. Thousands of years ago, it would have been even more diverse.
Thus, the map shows us that the Persian Empire was huge and was, therefore, sure to be diverse and cosmopolitan.