Why is south Korea a country?

South Korea is a country because at one point both North and South Korea were once Korea. The Korean war (1950-1953) resulted in the split. They both signed a treaty but they have major defences set up at the border. It is nearly impossible to escape from North to South (South- North) Korea. Out of what I know only one person has managed to escape to South Korea, a few has escaped from North Korea to China. If the Korean war didn't happen at all Korea would still be one nation.