# Difference between revisions of "Open mapping"

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A mapping of one topological space into another under which the image of every open set is itself open. | A mapping of one topological space into another under which the image of every open set is itself open. | ||

− | Projections of topological products onto the factors are open mappings. Openness of a mapping can be interpreted as a form of continuity of its inverse many-valued mapping. A one-to-one continuous open mapping is a [[Homeomorphism|homeomorphism]]. In general topology, open mappings are used in the classification of spaces. The question of the behaviour of topological invariants under continuous open mappings is important. All spaces with the [[First axiom of countability|first axiom of countability]], and only they, are images of metric spaces under continuous open mappings. A [[Metrizable space|metrizable space]] which is the image of a complete metric space under a continuous open mapping is metrizable by a complete metric. If a [[Paracompact space|paracompact space]] is the image of a [[Complete metric space|complete metric space]] under a continuous open mapping, then it is metrizable. A countable-to-one continuous open mapping of compacta does not increase the dimensions. However, a | + | Projections of topological products onto the factors are open mappings. Openness of a mapping can be interpreted as a form of continuity of its inverse many-valued mapping. A one-to-one continuous open mapping is a [[Homeomorphism|homeomorphism]]. In general topology, open mappings are used in the classification of spaces. The question of the behaviour of topological invariants under continuous open mappings is important. All spaces with the [[First axiom of countability|first axiom of countability]], and only they, are images of metric spaces under continuous open mappings. A [[Metrizable space|metrizable space]] which is the image of a complete metric space under a continuous open mapping is metrizable by a complete metric. If a [[Paracompact space|paracompact space]] is the image of a [[Complete metric space|complete metric space]] under a continuous open mapping, then it is metrizable. A countable-to-one continuous open mapping of compacta does not increase the dimensions. However, a $ 3 $- |

+ | dimensional cube can be mapped by a continuous open mapping onto a cube of any larger dimension. Every compactum is the image of a certain one-dimensional compactum under a continuous open mapping with zero-dimensional fibres (i.e. inverse images of points) | ||

Continuous open mappings under which the inverse images of all points are compact — the so-called compact-open mappings — are of separate interest in their own right. Spaces with a uniform base, and only they, are inverse images of metric spaces under compact-open mappings. Closed continuous open mappings are also important. All continuous open mappings of compacta into Hausdorff spaces (cf. [[Hausdorff space|Hausdorff space]]) fall into this category. Continuous closed open mappings preserve metrizability. Open mappings with discrete fibres play an important role in the theory of functions of one complex variable: these include all holomorphic functions in a domain. The theorem on the openness of holomorphic functions is central to proving the maximum-modulus principle, and to proving the fundamental theorem on the existence of a root of an arbitrary non-constant polynomial over the field of complex numbers. | Continuous open mappings under which the inverse images of all points are compact — the so-called compact-open mappings — are of separate interest in their own right. Spaces with a uniform base, and only they, are inverse images of metric spaces under compact-open mappings. Closed continuous open mappings are also important. All continuous open mappings of compacta into Hausdorff spaces (cf. [[Hausdorff space|Hausdorff space]]) fall into this category. Continuous closed open mappings preserve metrizability. Open mappings with discrete fibres play an important role in the theory of functions of one complex variable: these include all holomorphic functions in a domain. The theorem on the openness of holomorphic functions is central to proving the maximum-modulus principle, and to proving the fundamental theorem on the existence of a root of an arbitrary non-constant polynomial over the field of complex numbers. | ||

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====References==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1–2''' , Acad. Press (1966–1968) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.V. Keldysh, "Open and monotone mappings of compacta" , ''Proc. 3-rd All-Union Math. Congress'' , '''3''' , Moscow (1958) pp. 368–372 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1–2''' , Acad. Press (1966–1968) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.V. Keldysh, "Open and monotone mappings of compacta" , ''Proc. 3-rd All-Union Math. Congress'' , '''3''' , Moscow (1958) pp. 368–372 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD></TR></table> | ||

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====Comments==== | ====Comments==== | ||

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====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.T. Whyburn, "Topological analysis" , Princeton Univ. Press (1964)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Engelking, "General topology" , Heldermann (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.T. Whyburn, "Topological analysis" , Princeton Univ. Press (1964)</TD></TR></table> |

## Latest revision as of 08:04, 6 June 2020

A mapping of one topological space into another under which the image of every open set is itself open.

Projections of topological products onto the factors are open mappings. Openness of a mapping can be interpreted as a form of continuity of its inverse many-valued mapping. A one-to-one continuous open mapping is a homeomorphism. In general topology, open mappings are used in the classification of spaces. The question of the behaviour of topological invariants under continuous open mappings is important. All spaces with the first axiom of countability, and only they, are images of metric spaces under continuous open mappings. A metrizable space which is the image of a complete metric space under a continuous open mapping is metrizable by a complete metric. If a paracompact space is the image of a complete metric space under a continuous open mapping, then it is metrizable. A countable-to-one continuous open mapping of compacta does not increase the dimensions. However, a $ 3 $- dimensional cube can be mapped by a continuous open mapping onto a cube of any larger dimension. Every compactum is the image of a certain one-dimensional compactum under a continuous open mapping with zero-dimensional fibres (i.e. inverse images of points)

Continuous open mappings under which the inverse images of all points are compact — the so-called compact-open mappings — are of separate interest in their own right. Spaces with a uniform base, and only they, are inverse images of metric spaces under compact-open mappings. Closed continuous open mappings are also important. All continuous open mappings of compacta into Hausdorff spaces (cf. Hausdorff space) fall into this category. Continuous closed open mappings preserve metrizability. Open mappings with discrete fibres play an important role in the theory of functions of one complex variable: these include all holomorphic functions in a domain. The theorem on the openness of holomorphic functions is central to proving the maximum-modulus principle, and to proving the fundamental theorem on the existence of a root of an arbitrary non-constant polynomial over the field of complex numbers.

#### References

[1] | K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French) |

[2] | L.V. Keldysh, "Open and monotone mappings of compacta" , Proc. 3-rd All-Union Math. Congress , 3 , Moscow (1958) pp. 368–372 (In Russian) |

[3] | S. Stoilov, "The theory of functions of a complex variable" , 1–2 , Moscow (1962) (In Russian; translated from Rumanian) |

#### Comments

#### References

[a1] | R. Engelking, "General topology" , Heldermann (1989) |

[a2] | G.T. Whyburn, "Topological analysis" , Princeton Univ. Press (1964) |

**How to Cite This Entry:**

Open mapping.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Open_mapping&oldid=48046