Japanese space

Definition

• Say that a point x of a space X is Japanese (or X is Japanese at the point x) if X has a closure-preserving local base at the point x. The space X will be called Japanese if it is Japanese at every x ∈ X.

Remark

• Every isolated point is Japanese.
• Every Polish space is Japanese.

Reference

• Dow, A. , Ramírez Martínez, R. and Tkachuk, V. V. , A glance at spaces with closure preserving local bases. , Topology Appl. 157 (2010), no. 3, 548--558.