Diagonal product of mappings

$ f _ \alpha : X \rightarrow Y _ \alpha $, $ \alpha \in {\mathcal A} $

The mapping $ f: X \rightarrow Y = \prod \{ {Y _ \alpha } : {\alpha \in {\mathcal A} } \} $ defined by the equation $ f ( x) = \{ f _ \alpha ( x) \} \in Y $. The diagonal product of mappings $ f _ \alpha $ satisfies, for any $ \alpha $, the relation $ f _ \alpha = \pi _ \alpha f $, where $ \pi _ \alpha $ denotes the projection of the product $ Y $ on the factor $ Y _ \alpha $. The diagonal product of continuous mappings is continuous. A family of mappings $ f _ \alpha : X \rightarrow Y _ \alpha $ of topological spaces is said to be partitioning if for any point $ x \in X $ and neighbourhood $ Ox $ of $ x $ there exist an index $ \alpha $ and an open subset $ U _ \alpha \subset Y _ \alpha $ such that $ x \in f _ \alpha ^ { - 1 } U _ \alpha \subset Ox $. If $ \{ f _ \alpha : X \rightarrow Y _ \alpha \} $ is a partitioning family of mappings and if $ f $ is the diagonal product of the mappings $ f _ \alpha $, then $ f $ is an imbedding of $ X $ into the product $ \prod Y _ \alpha $, i.e. $ f: X \rightarrow fX $ is a homeomorphism. The diagonal product of mappings was used by A.N. Tikhonov to imbed a completely-regular space of weight $ \tau $ in the cube $ I ^ { \tau } $.

Comments

Instead of calling a family of mappings partitioning, one says that it separates points and closed sets.

In an arbitrary category with products, cf. Direct product, the diagonal product of mappings is given by the universal property defining the direct product. Indeed, categorically the product $ Y = \prod _ \alpha Y _ \alpha $ is an object together with morphisms: $ \pi _ \alpha : Y \rightarrow Y _ \alpha $ such that for every family of morphisms $ \phi _ \alpha : X \rightarrow Y _ \alpha $ there is a unique morphism $ f : X \rightarrow Y $ such that $ \pi _ \alpha f = f _ \alpha $.

Tikhonov's imbedding result is in [a2]. E. Čech, inspired by Tikhonov's result, obtained the following imbedding theorem [a1]: Let $ {\mathcal C} $ be the family of continuous mappings from a completely-regular space $ X $ into the unit interval $ I $. Then the diagonal mapping $ F: X \rightarrow I ^ {\mathcal C} $ is an imbedding, and the closure of $ F ( X) $ in $ I ^ {\mathcal C} $ is equivalent to the Stone–Čech compactification of $ X $.

References

[a1]	E. Čech, "On bicompact spaces" Ann. of Math. , 38 (1937) pp. 823–844 Zbl 63.0570.02
[a2]	A.N. [A.N. Tikhonov] Tichonoff, "Ueber die topologische Erweiterung von Räumen" Math. Ann. , 102 (1929) pp. 544–561