Mappings, classes of

The most important are classes of continuous mappings (cf. Continuous mapping), examined in general topology and its applications. These include: open mappings (the image of any open set is an open set, cf. Open mapping); closed mappings (the image of every closed set is closed, cf. Closed mapping); compact mappings (the inverse image of any point is a compact set, cf. Compact mapping); and perfect mappings (closed compact mappings, cf. Perfect mapping). Quotient mappings are defined by the requirement that a set is open in its image if and only if its complete inverse image is open (cf. Quotient mapping). Other important mappings are open compact mappings, pseudo-open mappings, contractions and condensing mappings. The latter are defined as one-to-one continuous mappings onto. Thus, in the classification of mappings in general topology, restrictions are imposed either on the behaviour of open or closed sets (during transition to the image), or on the properties of the inverse images of the sets. The second approach leads to the following classes of mappings in particular. Monotone mappings (the inverse image of every point is connected). Finite-to-one mappings are characterized by the fact that all inverse images of points are finite. Mappings under which the inverse image of every compact set is compact are called $ k $- mappings. By combining the restrictions of the first and second type, basic classes of continuous mappings in general topology can be singled out. By their very definitions, classes of mappings arrange themselves naturally into a sort of hierarchy, which may form a basis for a systematic classification of topological spaces [1]. This classification results from the solution of the following two types of questions. Let a class of spaces $ {\mathcal A} $ be given in which it is useful to differentiate, and let $ {\mathcal B} $ be a class of mappings from the original hierarchy. The images of spaces from $ {\mathcal A} $ have to be characterized by means of internal topological invariants under all possible mappings from $ {\mathcal B} $. Questions of the second type are analogous — the inverse images of the spaces from $ {\mathcal A} $ have to be characterized under mappings from $ {\mathcal B} $.

In solving these two types of question, theorems of a general nature which are far from obvious have been obtained. For example, spaces with the first axiom of countability are images of metric spaces (cf. Metric space) under continuous open mappings, and only they. Spaces with a uniform base (cf. Uniform topology), and only they, are images of metric spaces under open compact mappings. Fréchet–Urysohn spaces are pseudo-open images of metric spaces, while sequential spaces (cf. Sequential space) are quotient spaces of metric spaces. Moreover, inverse images of metric spaces under perfect mappings are paracompact feathered spaces (cf. Feathered space; Paracompact space), while inverse images of complete metric spaces (cf. Complete metric space) are Čech-complete paracompact spaces. Continuous images of spaces with a countable base are spaces with a countable net. In systematic research in this area, a single reciprocal classification of spaces and mappings has been obtained.

One class of continuous mappings which is of particular importance is the class of quotient mappings. The most important peculiarity of quotient mappings is that they provide a means of constructing new topological spaces. For instance, if a mapping $ f $ of a topological space $ X $ onto a set $ Y $ is given (for example, if the natural mapping $ \pi $ of a space $ X $ onto the set of all elements of a certain partition of this space is considered), then a natural topology can always be introduced on the set $ Y $, with the requirement that the mapping $ f $ be a quotient mapping: A set $ V \subset Y $ is open if and only if its complete inverse image $ f ^ { - 1 } ( V) $ is open in the space $ X $. Apart from those already mentioned, the most important are irreducible mappings (cf. Irreducible mapping), for example, in the theory of absolutes (cf. Absolute). See also Bifactorial mapping; Multi-valued mapping.

References

[1]	A.V. Arkhangel'skii, "Mappings and spaces" Russian Math. Surveys , 21 : 4 (1966) pp. 115–162 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 133–184

Comments

Related to monotone mappings (inverses of points are connected) one has: zero-dimensional mappings (inverses of points are zero-dimensional, cf. Zero-dimensional mapping) and light mappings (inverses of points are hereditarily disconnected). Quotient mappings (and spaces) are sometimes called factorial mappings (respectively, factor spaces).

A continuous mapping $ f : X \rightarrow Y $ is pseudo-open if for every $ y \in Y $ and open neighbourhood $ U $ of the set $ f ^ { - 1 } ( y) $ in $ X $ one has that $ y $ is in the interior of the set $ f ( U) \subset Y $.

References

[a1]	E.A. Michael, "A quintuple quotient quest" Gen. Topol. Appl. , 2 (1972) pp. 91–138
[a2]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)