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The People's Republic of China claims Taiwan as their 23rd province, though they have never controlled Taiwan or the present Republic of China area.
The Republic of China is commonly known as Taiwan. The People's Republic of China is commonly known simply as China.
The People's Republic of China also claims the island of Taiwan to be a part of its territory. Taiwan is officially known as the Republic of China.
China itself is just a region. The countries are the People's Republic of China and, if you agree, the Republic of China (Taiwan).
they fled to the island of Taiwan, where they established the Republic of China.