A topological space (X, tau) is called submetric if tau contains a metric topology tau_m, i.e., there exists a continuous metric d on X generating a topology tau_d that is weaker than tau (tau_d subset tau). The name was coined by Jakubowski (2023), building on his 1997 work on non-metrizable topologies.
The prototypical example is a separable Hilbert space H with the weak topology tau_w: the norm metric is continuous w.r.t. tau_w (since the norm is weakly lower semicontinuous and weak convergence implies norm convergence of inner products), but tau_w is strictly weaker than the norm topology on infinite-dimensional H. In particular, (H, tau_w) is NOT metrizable (the weak topology on the unit ball is metrizable, but not on all of H), yet it IS submetric.
The key property of submetric spaces is that tau-compact sets are metrizable: if K is tau-compact, then the identity (K, tau) → (K, tau_d) is a continuous bijection from a compact space to a Hausdorff space, hence a homeomorphism. This means tau and tau_d coincide on K. Consequently, sequences in tau-compact sets behave exactly as in metric spaces.
This property enables a sequential Prokhorov theorem (Theorem 3.1/4.2 in Jakubowski 2023): on a submetric space, every uniformly tau-tight family of probability measures has the property that every sequence contains a subsequence admitting a Skorokhod representation. This recovers the compactness half of Prokhorov’s theorem without requiring global metrizability.
A submetric space automatically has property CCSP (Countable family of Continuous functions Separating Points): the metric d can be written as d(x,y) = sum 2^{-i} |f_i(x) - f_i(y)| / (1 + |f_i(x) - f_i(y)|) for some countable family {f_i}. The family {f_i} separates points and is continuous w.r.t. tau.
Key Details
- Definition (Jakubowski 2023, Def 4.1): (X, tau) is submetric if tau contains a metric topology tau_m
- Examples: separable Banach spaces with weak topology; Skorokhod space D with S-topology; dual spaces with weak* topology
- Compact sets are metrizable: tau and tau_d agree on tau-compact subsets
- Sequential Prokhorov: uniform tau-tightness implies sequential compactness (subsequences with Skorokhod representations)
- Tightness is essential: Remark 3.3 in Jakubowski (2023) — spaces with CCSP need not be Prohorov; uniform tightness cannot be dropped
- Fernique’s example: in (H, tau_w), sequences can converge in distribution yet fail to be uniformly tau_w-tight, showing tau-tightness is genuinely stronger than d-boundedness