Totally Bounded

Some articles on totally bounded, totally:

Glossary Of Topology - T
... Totally bounded A metric space M is totally bounded if, for every r > 0, there exist a finite cover of M by open balls of radius r ... compact if and only if it is complete and totally bounded ... Totally disconnected A space is totally disconnected if it has no connected subset with more than one point ...
Uniform Property - Uniform Properties
... Totally bounded (or Precompact) ... A uniform space X is totally bounded if for each entourage E ⊂ X × X there is a finite cover {Ui} of X such that Ui × Ui is contained in E for all i ... Equivalently, X is totally bounded if for each entourage E there exists a finite subset {xi} of X such that X is the union of all E ...
Totally Bounded Space - Use of The Axiom of Choice
... 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 ... (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 ...

Famous quotes containing the words bounded and/or totally:

    “I could be bounded in a nutshell and count myself a king of
    infinite space, were it not that I have bad dreams.”
    —William Shakespeare (1564–1616)

    “Postmodernism is, almost by definition, a transitional cusp of social, cultural, economic and ideological history when modernism’s high-minded principles and preoccupations have ceased to function, but before they have been replaced with a totally new system of values. It represents a moment of suspension before the batteries are recharged for the new millennium, an acknowledgment that preceding the future is a strange and hybrid interregnum that might be called the last gasp of the past.”
    —Gilbert Adair, British author, critic. Sunday Times: Books (London, April 21, 1991)