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Should this article be expanded to reflect the "ongoing space race" engaged in by Asian and other nations? (No.)
No, for exactly the same reason that American Civil War should not be expanded to include the 1960-1996 Guatemalan Civil War. The scope of this article concerns an era of history during the Cold War when the United States competed with the Soviet Union for preeminence in space exploration. It ended by the time of the Soviet Union's collapse and the end of the Cold War. Subsequent space programs developed by other nations therefore cannot be a part of it, despite journalists' attempts to re-use the term. Wikipedia should cover topics in their proper historical perspective, and their timeless facets recognized by Wikipedia consensus should not be muddled or diffused by recent events; see WP:Recentism, Latter-day space programs should be covered in other articles.
Shouldn't this article say that the Soviet Union or the United States won the space race? (No.)
No, for the following reasons:
  • It was not literally a "race", with the sole objective being to achieve milestones first. It was a competition with the goal of demonstrating mastery of spaceflight technology. While the USSR appeared to "win" early on by being the first to launch a satellite, a human, solo flights of several days duration, the first multi-person crew, and the first spacewalk, the US was soon able to do the same things, and this did not actually prove that one nation's technology was superior to the other's.
  • The USSR soon lost this lead because they had cut corners to turn Vostok into Voskhod and had to cancel this program after only two flights. Meanwhile, the US Gemini program caught up with long duration flights, the ability to rendezvous and dock two craft in orbit, and demonstrating the ability to do useful work outside the spacecraft without tiring. By any objective measure, this switched the advantage to the US.
  • The USSR made the mistake of not developing either large rocket engines or more efficient liquid hydrogen-fueled rockets until much later than the US, and therefore failed to develop a super-heavy lift launch vehicle required for human lunar flight before the US. The US met its goal of landing humans safely on the Moon in 1969, which historians recognize worldwide as a superior achievement to those of the USSR.
  • Conversely, despite American achievements on the Moon, the USSR was still able to launch the first space station. In essence, the space "race" concluded with a "tie," as both nations achieved feats of technological mastery and exploration that significantly improved humanity's understanding of the cosmos.