A convergence not metrizable

Certain notions of convergence of sequences functions such as pointwise convergence and (uniform) convergence on compact or bounded sets come from suitable topological function spaces; see [1]. Under certain conditions these topologies involved are metrizable, which in an advantage since there is an extensive theory on convergence in metric spaces. However, the case of pointwise convergence is delicate, since it is shown that under certain hypotheses this form of convergence of sequences of functions is not equivalent to convergence in metric.

💡 Research Summary

The paper addresses the long‑standing question of whether pointwise convergence of sequences of functions can be captured by a metric on the underlying function space. The author works in a fairly general setting: let ((M,d_M)) be a complete metric space that is “strongly second countable”, meaning there exists a countable dense subset (D\subset M) whose complement (M\setminus D) is also dense. Typical examples are (\mathbb R) with its usual topology, where (D=\mathbb Q). Let ((N,d_N)) be any metric space that possesses a non‑trivial path component (i.e., there are distinct points that can be joined by a continuous path). The main claim (Proposition 0.2) is that under these hypotheses there is no metric (d) on the full function space (F(M,N)={f:M\to N}) such that convergence with respect to (d) coincides with pointwise convergence.

The proof proceeds by contradiction. Assuming such a metric (d) exists, the author exploits the strong second‑countability of (M) to pick a dense countable set (D) and its dense complement. Choose two distinct points (a,b\in N) and a continuous path (\psi: