Final answer:
Secretary of State John Hay convinced European powers to maintain an open-door policy in China after the Boxer Rebellion, preventing the division of China into colonies and ensuring all nations could trade freely.
Step-by-step explanation:
After the Boxer Rebellion, Secretary of State John Hay convinced European powers not to divide China into colonies. In the wake of rising attempts by European nations and Japan to carve out spheres of influence in China, Hay introduced the Open Door Notes in 1899. These notes proposed that all nations be allowed to trade freely in China without interference within the spheres.
Although the idea was initially met with mixed responses, the violent outburst of the Boxer Rebellion spurred an international coalition to quell the uprising and maintain foreign trading rights in China. Post-rebellion, Secretary Hay advocated the Open Door policy, essentially committed to respecting China's territorial integrity while enabling free trade for all powers.