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Totally bounded space

A Wisdom Archive on Totally bounded space

Totally bounded space

A selection of articles related to Totally bounded space

More material related to Totally Bounded Space can be found here:
Index of Articles
related to
Totally Bounded Space
Totally bounded space, Totally bounded space - Definitions, Totally bounded space - Examples and nonexamples, Totally bounded space - Relationships with compactness and completeness, Totally bounded space - Use of the axiom of choice

ARTICLES RELATED TO Totally bounded space

Totally bounded space: Encyclopedia II - Totally bounded space - Definitions

The general logical form of the definition is: A subset S of a space X is a totally bounded set if and only if, given any size E, there exist a natural number n and a family A1, A2, ..., An of subsets of X, such that S is contained in the union of the family (in other words, the family is a finite cover of S), and such that each set Ai in the family is of size E (or ...

See also:

Totally bounded space, Totally bounded space - Definitions, Totally bounded space - Examples and nonexamples, Totally bounded space - Relationships with compactness and completeness, Totally bounded space - Use of the axiom of choice

Read more here: » Totally bounded space: Encyclopedia II - Totally bounded space - Definitions

Totally bounded space: Encyclopedia II - Totally bounded space - Relationships with compactness and completeness

There is a nice relationship between total boundedness and compactness: A space is compact if and only if it is both totally bounded and Cauchy complete. This can be seen as a generalisation of the Heine-Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness (and also replace closedness with completeness). There is a complementary relationship between total boundedness and the process of Cauchy completion: A space is totally bounded if and only if its Cauchy completion is totally bounded. (This corresponds to the ...

See also:

Totally bounded space, Totally bounded space - Definitions, Totally bounded space - Examples and nonexamples, Totally bounded space - Relationships with compactness and completeness, Totally bounded space - Use of the axiom of choice

Read more here: » Totally bounded space: Encyclopedia II - Totally bounded space - Relationships with compactness and completeness

More material related to Totally Bounded Space can be found here:
Index of Articles
related to
Totally Bounded Space
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