View source for Continuous decomposition

''of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256101.png" />''

A covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256102.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256103.png" /> by pairwise-disjoint non-empty sets satisfying the following condition: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256104.png" /> and any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256105.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256106.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256107.png" /> there is a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256108.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c0256109.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561010.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561011.png" /> that is the union of a certain collection of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561012.png" />. A decomposition is continuous if and only if the corresponding [[Quotient mapping|quotient mapping]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561013.png" /> onto the space of this decomposition is closed. A continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561014.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561015.png" /> onto a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561016.png" /> is closed if and only if the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561018.png" /> is continuous.

Continuous decompositions occur frequently in the theory of compact spaces (cf. [[Compact space|Compact space]]). Every continuous mapping of a compact space onto a Hausdorff space is closed. Therefore, every continuous mapping of a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561019.png" /> onto a Hausdorff space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561020.png" /> gives rise to a continuous decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561021.png" /> 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|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|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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561022.png" />-dimensional projective space, from the topological point of view, is the space of the continuous decomposition of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561023.png" />-dimensional sphere, lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561024.png" />-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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====
The space of a decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561026.png" /> has the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561027.png" /> as underlying set, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561028.png" /> is open if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561029.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561030.png" />. That is, it is the quotient space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561031.png" /> defined by the equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561032.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561034.png" /> both belong to the same element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561035.png" />.

A covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561036.png" /> satisfying the requirements of the article above is also called an upper semi-continuous decomposition. An equivalent definition is: For every closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561037.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561038.png" /> is closed. A covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561039.png" /> is called a lower semi-continuous decomposition if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561040.png" /> is open for every open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561041.png" />. Equivalently: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025610/c02561042.png" /> is a lower semi-continuous decomposition if and only if the corresponding quotient mapping is open (cf. [[Open mapping|Open mapping]]).

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