The key for Black City in Pokemon White 2.
This should help you. http://www.ign.com/wikis/pokemon-black-white-version-2/Key_System
You have to defeat the 5th area in Black tower. That gives you the key to Black City.
Source: http://www.marriland.com/black2_white2/walkthrough/white-forest-black-city/4239-black-tower
http://www.ign.com/wikis/pokemon-black-white-version-2/Key_System
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