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)
“By choice they made themselves immune
To pity and whatever moans in man
Before the last sea and the hapless stars;
Whatever mourns when many leave these shores;
Whatever shares
The eternal reciprocity of tears.”
—Wilfred Owen (1893–1918)