Open figure

Open mapping is a term used in mathematics , especially topology .

Continuous functions can be characterized in that archetypes of open subsets of the target set are open again. The correspondingly formulated condition for images instead of archetypes leads to the concept of open mapping .

definition

A mapping (or function) from a topological space into a topological space is called open if the image of each open subset of is an open subset of . ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f(O)}$ ${\displaystyle O}$ ${\displaystyle X}$ ${\displaystyle Y}$

Explanations and examples

• A mapping is open if and only if there is a neighborhood of in for every point and every neighborhood of in in the image set . ${\displaystyle f\colon \,X\to Y}$ ${\displaystyle x\in X}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle f(U)}$ ${\displaystyle f(x)}$ ${\displaystyle Y}$
• If , and are topological spaces and the images and both are open, then the composition is also an open image. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle f\colon \,X\to Y}$ ${\displaystyle g\colon \,Y\to Z}$ ${\displaystyle g\circ f\colon X\to Z}$
• Open mappings are usually not continuous. For example, the mapping defined by and according to Picard's great theorem is an open map, but not continuous in . ${\displaystyle f(0)=0}$ ${\displaystyle f(z)=e^{\frac {1}{z}}}$ ${\displaystyle 0}$
• An example of a continuous and non-open mapping is the mapping defined by an irrational number . The image in this figure is not an open subset, but lies close in . ${\displaystyle \alpha }$ ${\displaystyle f(x)=(e^{x2\pi i},e^{\alpha x2\pi i})}$ ${\displaystyle f\colon \mathbb {R} \to S^{1}\times S^{1}}$ ${\displaystyle S^{1}\times S^{1}}$
• If there is a discrete topological space , then every mapping to is an open mapping, but only the locally constant maps are continuous. ${\displaystyle Y}$ ${\displaystyle Y}$
• Continuous images are usually not open. As a rule, a constant mapping is not open. The same example shows that closed figures do not have to be open.
• The mapping defined by is also continuous, but not open because it is not open in . ${\displaystyle f(x)=x^{2}}$ ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle f((-1,1))=[0,1)}$ ${\displaystyle \mathbb {R} }$
• An open mapping is usually not completed. The mapping is open, the image set of the closed set is the non-closed set . ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ,\,(s,t)\mapsto s}$ ${\displaystyle \{(s,t)\colon s\geq 0,st\geq 1\}}$ ${\displaystyle (0,\infty )}$
• Homeomorphisms are always open mappings. A continuous bijection is a homeomorphism if and only if it is an open map.
• If and are topological spaces and is a bijection, then is a homeomorphism if and only if both and the inverse mapping are open mappings. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon \,X\to Y}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f^{-1}\colon \,Y\to X}$
• In a topological product space , the canonical projections are always open. ${\displaystyle X=\textstyle \prod _{i\in I}X_{i}}$ ${\displaystyle p_{i}\colon X\to X_{i}}$
• Projections of fiber bundles are always open images.
• A continuous, linear operator between two Banach spaces is open if and only if it is surjective . ( Sentence about the open figure )
• The openness theorem of function theory says that holomorphic functions that are not constant on any connected component of their open domain are open mappings.
• According to the theorem of the invariance of open sets , in Euclidean space the image set is always open for every open subset and every injective continuous mapping , i.e. an open mapping. ${\displaystyle \mathbb {R} ^{n}\;(n\in \mathbb {N} )}$ ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ ${\displaystyle f\colon U\to \mathbb {R} ^{n}}$ ${\displaystyle f(U)\subseteq \mathbb {R} ^{n}}$ ${\displaystyle f}$
• The theorem of the open mapping from the theory of locally compact groups says that every continuous, surjective group homomorphism from a σ-compact, locally compact group to a locally compact group is automatically open.

literature

• Nicolas Bourbaki : General Topology. Chapters 1-4. Springer, Berlin (below) 1998, ISBN 978-3-540-64241-1
• Wolfgang Franz : Topology I. Walter de Gruyter, 1973, ISBN 3-11-004117-0
• Egbert Harzheim : Introduction to Combinatorial Topology Scientific Book Society, Darmstadt 1978, ISBN = 3-534-07016-X
• Joseph Muscat : Functional Analysis. An introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. Springer, Cham (u. U.) 2014, ISBN 978-3-319-06727-8
• Boto von Querenburg : Set theoretical topology. 3. Edition. Springer, Berlin 2001, ISBN 3-540-67790-9
• Horst Schubert : Topology. 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6

Individual evidence

1. ↑ Dirk Werner : Functional Analysis. 6th edition. Springer-Verlag, 2007, ISBN 978-3-540-72533-6 , behind definition IV.3.1.