Totally Bounded Space - Use of The Axiom of Choice

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 choice and/or axiom:

    The question of place and climate is most closely related to the question of nutrition. Nobody is free to live everywhere; and whoever has to solve great problems that challenge all his strength actually has a very restricted choice in this matter. The influence of climate on our metabolism, its retardation, its acceleration, goes so far that a mistaken choice of place and climate can not only estrange a man from his task but can actually keep it from him: he never gets to see it.
    Friedrich Nietzsche (1844–1900)

    It’s an old axiom of mine: marry your enemies and behead your friends.
    —Robert N. Lee. Rowland V. Lee. King Edward IV (Ian Hunter)