The hemispheres of Magdeburg

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The atmospherical air-pressure outside of the sphere is about 1 atm. If the outlet of the sphere is open, the pressure inside of the sphere is just the same. The forces that affect the inside and the outside of the sphere are balanced. (see image below)

When the air is removed from the sphere and the outlet is closed there is a vacuum inside of the sphere. Because there is (almost) no air left inside of the sphere and therefore the air-pressure close to 0, (almost) no force affects the inside of the sphere. Outside, nothing has changed. There is still a pressure of 1 atm that is affecting the outside of the sphere. (see image below)
This pressure presses together the two hemispheres. In order to separate them, a bigger force has to be dedicated than the force that keeps the hemispheres together.