Actual figure
An actual mapping is a continuous mapping that is examined in set theoretical topology , a branch of mathematics.
Definitions
The definition of actual images varies from author to author. Two common definitions are therefore presented here.
- A continuous mapping between two locally compact spaces actually means that the archetype of every compact set is compact .
Another and more general definition is:
- A continuous mapping between two topological spaces actually means exactly when the mapping is closed for any topological space Z.
The second definition is equivalent to the first if there is a Hausdorff space and a locally compact Hausdorff space.
Examples
- If the definition set is compact, then every continuous mapping is actually.
- Every homeomorphism is actually, including every diffeomorphism and every biholomorphic mapping .
properties
- An actual mapping is completed , that is, the image of each completed set is completed.
- The restriction of the actual mapping to a closed subspace is always real.
- The composition of actual images is real again. Topological spaces together with the actual mappings thus form a sub-category of the category of continuous functions.
- If topological spaces are actual mappings, then it is again an actual map.
- If there is an actual mapping between topological spaces and is compact , then is compact in .
- If a compact space and any topological space and the topological product, then the projection is an actual mapping.
Applications
Actual maps provide a criterion for the compactness of a topological space: Let it be a one-element topological space with the only existing topology. Then the following applies: A topological space is compact if and only if the constant mapping is actually. The last two properties mentioned follow from this.
literature
- Gerd Laures, Markus Szymik: Basic Topology Course. Spektrum Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2040-4 .
- Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (= Berlin study series on mathematics. Vol. 15). Heldermann, Lemgo 2006, ISBN 3-88538-115-X .
- Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer-Verlag, Berlin et al. 2001, ISBN 3-540-67790-9 .