Continuous decomposition

of a topological space

A covering  of  by pairwise-disjoint non-empty sets satisfying the following condition: For any  and any neighbourhood  of  in  there is a neighbourhood  of  in  contained in  that is the union of a certain collection of elements of . A decomposition is continuous if and only if the corresponding quotient mapping of  onto the space of this decomposition is closed. A continuous mapping  of a space  onto a space  is closed if and only if the decomposition  of  is continuous.

Continuous decompositions occur frequently in the theory of compact spaces (cf. Compact space). Every continuous mapping of a compact space onto a Hausdorff space is closed. Therefore, every continuous mapping of a compact space  onto a Hausdorff space  gives rise to a continuous decomposition of  by inverse images of points. For the same reason a decomposition of a Hausdorff compactum is continuous if (and only if) the space of this decomposition satisfies the Hausdorff separation axiom. The merit of a continuous decomposition of a space into closed sets is the preservation of normality and paracompactness. By way of contrast, the space of a continuous decomposition of a metric space into closed sets need not be metrizable. The simplest example of this situation is the space of the decomposition of the plane whose only non-trivial element is a fixed straight line.

Like decomposition in general, continuous decompositions are an important tool in the construction of new topological spaces from existing ones and also in the representation of more complicated topological spaces as the spaces of continuous decompositions of simpler or more standard spaces. Thus, every metrizable compactum is the space of a continuous decomposition of a Cantor set. Every connected, locally connected, metrizable compactum can be represented as the space of a continuous decomposition of an interval. Continuous decompositions occur naturally in certain specific constructions; for example, the projective plane, regarded as a topological space, is the space of the continuous decomposition of the ordinary sphere into pairs of diametrically-opposite points. Similarly, -dimensional projective space, from the topological point of view, is the space of the continuous decomposition of the -dimensional sphere, lying in -dimensional Euclidean space, into pairs of diametrically-opposite points. In this language, a Möbius strip can be described accurately as the space of a certain continuous decomposition of a rectangle, and other geometric objects can also be constructed in this way.

#### References[edit]

[1] |  A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

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#### Comments[edit]

The space of a decomposition  of  has the set  as underlying set, and  is open if and only if  is open in . That is, it is the quotient space of  defined by the equivalence relation  if and only if  and  both belong to the same element of .

A covering  satisfying the requirements of the article above is also called an upper semi-continuous decomposition. An equivalent definition is: For every closed set  the set  is closed. A covering  is called a lower semi-continuous decomposition if  is open for every open set . Equivalently:  is a lower semi-continuous decomposition if and only if the corresponding quotient mapping is open (cf. Open mapping).

Instead of "decomposition" the word "partitioning" is used.