Use of The Axiom of Choice
The properties of total boundedness mentioned above rely in part on the axiom of choice. In the absence of the axiom of choice, total boundedness and precompactness must be distinguished. That is, we define total boundedness in elementary terms but define precompactness in terms of compactness and Cauchy completion. It remains true (that is, the proof does not require choice) that every precompact space is totally bounded; in other words, if the completion of a space is compact, then that space is totally bounded. But it is no longer true (that is, the proof requires choice) that every totally bounded space is precompact; in other words, the completion of a totally bounded space might not be compact in the absence of choice.
Read more about this topic: Totally Bounded Space
Famous quotes containing the words axiom and/or choice:
“It is an axiom in political science that unless a people are educated and enlightened it is idle to expect the continuance of civil liberty or the capacity for self-government.”
—Texas Declaration of Independence (March 2, 1836)
“Then did they strive with emulation who should repeat most wise maxims importing the necessity of suspicion in the choice of our friends—such as “mistrust is the mother of security,” with many more to the same effect.... But notwithstanding the esteem which they professed for suspicion, yet did they think proper to veil it under the name of caution.”
—Sarah Fielding (1710–1768)