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The beginning of "outer space" doesn't have a strict formal definition. The layers of the atmosphere are shown in a number of online diagrams (see http://ds9.ssl.berkeley.edu/LWS_GEMS/3/graph_1.htm for example). From the top of Everest at about 8850 m it is only a couple of km to the tropopause (troposphere/stratosphere boundary) - the exact height of which varies by both latitude and season. But the outermost part of the atmosphere, the exosphere, is interpreted as extending from about 500 to 10,000 or even 60,000 km above the Earth's surface. So - it is a long, long way! Finally - viewed in another way outer space is much closer. The atmospheric pressure at the summit of Everesit is only about 30% of that at sea level. That is, about 70% of the mass of the atmosphere lies beneath you. [That's why bottlled oxygen is commonly used for climbers.] At an elevation of 50 km or so the pressure is a small fraction of 1%.
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