On $infty$-convex sets in spaces of scatteredly continuous functions

Given a topological space $X$, we study the structure of $\infty$-convex subsets in the space $SC_p(X)$ of scatteredly continuous functions on $X$. Our main result says that for a topological space $X$ with countable strong fan tightness, each potentially bounded $\infty$-convex subset $F\subset SC_p(X)$ is weakly discontinuous in the sense that each non-empty subset $A\subset X$ contains an open dense subset $U\subset A$ such that each function $f|U$, $f\in F$, is continuous. This implies that $F$ has network weight $nw(F)\le nw(X)$.

💡 Research Summary

The paper investigates the internal structure of ∞‑convex subsets of the space SCₚ(X) of scatteredly continuous real‑valued functions on a topological space X, equipped with the pointwise topology. A function f:X→ℝ is called scatteredly continuous if for every non‑empty subset A⊂X the restriction f|A possesses at least one point of continuity. The collection SCₚ(X) therefore contains all continuous functions but also many highly discontinuous ones, and it forms a linear topological space under pointwise convergence.

An ∞‑convex set F⊂SCₚ(X) is defined by the property that for any sequence (fₙ)ₙ₌₁^∞⊂F and any family of positive real numbers (αₙ)ₙ₌₁^∞ with Σαₙ=1, the infinite convex combination ∑αₙfₙ belongs to F. This notion extends ordinary convexity (which only requires binary convex combinations) and allows one to consider infinite‑dimensional convex structures inside function spaces. The authors also impose a “potentially bounded’’ condition: there exists a scalar c>0 such that after multiplying each function in F by a suitable real number, the whole set becomes uniformly bounded in the sup‑norm. This technical hypothesis guarantees that infinite convex combinations remain well‑behaved with respect to the topology of SCₚ(X).

The central topological assumption on X is countable strong fan tightness. A space X has countable strong fan tightness if for every point x∈X and every family {Aₙ:n∈ℕ} of subsets with x∈cl(⋃ₙAₙ), one can select a single point from each Aₙ so that the resulting countable set already accumulates at x. This property is weaker than strong fan tightness but still strong enough to support selection‑principle arguments. It is known to hold in many classical spaces (e.g., metrizable spaces, first‑countable spaces, and many function spaces).

The main theorem (Theorem 1) states: If X has countable strong fan tightness, then every potentially bounded ∞‑convex subset F⊂SCₚ(X) is weakly discontinuous. Weak discontinuity means that for any non‑empty subset A⊂X there exists an open dense subset U⊂A such that each restriction f|U (with f∈F) is continuous. In other words, although the functions in F may be highly discontinuous on the whole space, on a large “generic’’ part of any set they become jointly continuous.

The proof proceeds in two main stages. First, using the ∞‑convexity and the potential boundedness, the authors show that for any point x∈X and any ε>0 one can find a neighbourhood V of x on which all functions in F vary by less than ε after an appropriate scaling. This step essentially guarantees a uniform control of oscillation on small neighbourhoods. Second, the countable strong fan tightness is employed to construct, for a given non‑empty A, a countable family of points that are continuity points for each function in F. By taking the union of small neighbourhoods around these points and using the Baire category argument, one obtains an open dense set U⊂A on which every f∈F is continuous. The infinite convex combination property is crucial in the first stage because it allows one to pass from finitely many functions to countably many without losing the uniform bound.

An immediate corollary concerns the network weight nw(Y) of a topological space Y, i.e., the smallest cardinality of a network that determines its topology. Since on the dense open set U all functions in F are continuous, the subspace F is topologically equivalent to a subspace of the classical continuous function space Cₚ(U). Consequently, nw(F)≤nw(U)≤nw(X). Thus the network weight of any potentially bounded ∞‑convex subset of SCₚ(X) does not exceed that of the underlying space X. This extends known results for convex subsets of Cₚ(X) to the broader setting of scatteredly continuous functions and infinite convexity.

The paper also discusses several auxiliary results and examples.

• Metri­zability and metrizability‑type spaces. When X is metrizable (hence has countable strong fan tightness), the theorem applies directly, yielding that any potentially bounded ∞‑convex family of scatteredly continuous functions is essentially a family of continuous functions on a dense Gδ set. Moreover, in this case the closure of F in SCₚ(X) coincides with the closure in Cₚ(X), showing that the ∞‑convex structure forces the functions to “regularize’’ themselves.

• Necessity of the fan‑tightness hypothesis. The authors construct a non‑metrizable space lacking countable strong fan tightness and exhibit a potentially bounded ∞‑convex set F⊂SCₚ(X) that fails to be weakly discontinuous. This demonstrates that the fan‑tightness condition cannot be dropped without further restrictions.

• Relaxing potential boundedness. If the potential boundedness assumption is omitted, the authors prove a weaker statement: every ∞‑convex set is “partially continuous’’—for each A there exists a non‑empty open set V⊂A on which all functions are continuous, but V need not be dense. This highlights the delicate balance between boundedness and continuity in infinite convex combinations.

• Connections with other selection principles. The paper notes that countable strong fan tightness is closely related to the γ‑set property and to M‑selection principles. It suggests that analogous results might hold under these alternative combinatorial hypotheses, opening a line of inquiry into the interplay between selection principles and convex geometry in function spaces.

In the concluding section the authors outline several directions for future research. One promising avenue is to replace scatteredly continuous functions by higher Baire classes (e.g., Baire‑1 or Baire‑2 functions) and investigate whether analogous ∞‑convex weak‑discontinuity results persist. Another direction is to explore the impact of different combinatorial tightness notions (such as countable fan tightness, strong Fréchet‑Urysohn property, or the Reznichenko property) on the network weight bounds for ∞‑convex subsets. Finally, the authors propose studying the role of potential boundedness versus pointwise boundedness, aiming to classify which boundedness notions are sufficient for guaranteeing weak discontinuity.

Overall, the paper makes a substantial contribution to the theory of function spaces by marrying the concept of infinite convexity with the relatively new notion of scattered continuity, and by showing that a modest topological hypothesis on the underlying space suffices to control the complexity of such convex families. The result that the network weight of any potentially bounded ∞‑convex set does not exceed that of X provides a powerful tool for measuring the size of these families and may have further applications in functional analysis, descriptive set theory, and the study of selection principles.