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 . ${\ displaystyle f \ colon X \ to Y}$

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. ${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle f \ times \ operatorname {id} _ {Z} \ colon X \ times Z \ to Y \ times Z, (x, z) \ mapsto (f (x), z)}$

The second definition is equivalent to the first if there is a Hausdorff space and a locally compact Hausdorff space. ${\ displaystyle X}$ ${\ displaystyle Y}$

Examples

If the definition set is compact, then every continuous mapping is actually. ${\ displaystyle X}$ ${\ displaystyle f \ colon X \ to Y}$
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. ${\ displaystyle f | A \ colon A \ to Y}$ ${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle A \ subseteq X}$

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.

{\ displaystyle X_ {1}, X_ {2}, Y_ {1}, Y_ {2}}

{\ displaystyle f_ {1} \ colon X_ {1} \ to Y_ {1}, f_ {2} \ colon X_ {2} \ to Y_ {2}}

{\ displaystyle f_ {1} \ times f_ {2} \ colon X_ {1} \ times X_ {2} \ to Y_ {1} \ times Y_ {2}}

If there is an actual mapping between topological spaces and is compact , then is compact in . ${\ displaystyle f \ colon X \ to Y}$ ${\ displaystyle B \ subseteq Y}$ ${\ displaystyle f ^ {- 1} (B)}$ ${\ displaystyle X}$

If a compact space and any topological space and the topological product, then the projection is an actual mapping. ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle X \ times Y}$ ${\ displaystyle \ pi _ {Y} \ colon X \ times Y \ to Y, (x, y) \ mapsto y}$

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. ${\ displaystyle P}$ ${\ displaystyle X}$ ${\ displaystyle p \ colon X \ to P}$

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 .