**Complete Metric Space**

In mathematical analysis, a metric space *M* is called **complete** (or a **Cauchy space**) if every Cauchy sequence of points in *M* has a limit that is also in *M* or, alternatively, if every Cauchy sequence in *M* converges in *M*.

Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it. (See the examples below.) It is always possible to "fill all the holes", leading to the *completion* of a given space, as explained below.

Read more about Complete Metric Space: Examples, Some Theorems, Completion, Topologically Complete Spaces, Alternatives and Generalizations

### Other articles related to "metric, complete metric space, space, metrics, spaces, metric spaces":

... and to oppose the compulsory imposition of the

**Metric**system in the UK ... schoolchildren had been educated using only

**metric**measures since 1974 (earlier in some places), and British industry had changed to using

**metric**tools and equipment during the 1980's and were in most cases ...

... Metric (band), a Canadian indie rock band. ...

... BCT1 also shows that every

**complete metric space**with no isolated points is uncountable ... If X is a countable

**complete metric space**with no isolated points, then each singleton {x} in X is nowhere dense, and so X is of first category in itself.) In particular, this proves that the ... BCT1 shows that each of the following is a Baire

**space**The

**space**R of real numbers The irrational numbers, with the

**metric**defined by d(x, y) = 1 / (n + 1), where n ...

... A variety of different

**metrics**can be defined on a shift

**space**... One can define a

**metric**on a shift

**space**by considering two points to be "close" if they have many initial symbols in common this is the p-adic

**metric**... In fact, both the one- and two-sided shift

**spaces**are compact

**metric spaces**...

**Complete Metric Space**- Alternatives and Generalizations

... groups, an alternative to relying on a

**metric**structure for defining completeness and constructing the completion of a

**space**is to use a group structure ... This is most often seen in the context of topological vector

**spaces**, but requires only the existence of a continuous "subtraction" operation ... the distance between two points and is gauged not by a real number via the

**metric**in the comparison, but by an open neighbourhood of via subtraction in the comparison ...

### Famous quotes containing the words space and/or complete:

“The secret ones around a stone

Their lips withdrawn in meet surprise

Lie still, being naught but bone

With naught but *space* within their eyes....”

—Allen Tate (1899–1979)

“The history of any nation follows an undulatory course. In the trough of the wave we find more or less *complete* anarchy; but the crest is not more or less *complete* Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.”

—Aldous Huxley (1894–1963)