Tychonoffrom

Tychonoffrom is a hypothetical concept in topology introduced to generalize the role of function separation in product-compactness phenomena. It describes a topological space together with a chosen family of real-valued continuous functions that control how separation and compactness interact across product constructions.

Formally, a Tychonoffrom space is a pair (X, F) where X is a completely regular Hausdorff space and F is a subfamily of C(X, R) that separates points from closed sets. The separation condition means that for any x in X and closed set C with x not in C, there exists a function f in F with f(x) not equal to sup{f(y) : y in C}. The defining feature of a Tychonoffrom is the compactness-like condition imposed by the evaluation map e_F: X → R^F, defined by e_F(x) = (f(x))_{f in F}. The pair is called Tychonoffrom if the image e_F(X) is compact in the product topology on R^F.

Relation to standard concepts: If X is compact and F separates points from closed sets, then e_F(X) is compact, so (X, F) is a Tychonoffrom. When F is taken to be all continuous real-valued functions on X, the Tychonoffrom framework recovers the familiar setting of classical Tychonoff spaces, viewed through the lens of their full function spaces.

Products and stability: For a family {(X_i, F_i)} of Tychonoffrom spaces, one can form a product X = ∏ X_i with a corresponding family F generated by the coordinate functions. If each e_{F_i}(X_i) is compact, the product e_F(X) is compact, illustrating a product-stability aspect of the concept.

See also: Tychonoff theorem, completely regular spaces, product topology, compactness, C(X, R).