## Logic Seminar

A natural question to ask is given a space in the ground model, is there a natural interpretation of it in some generic extension? In general, this question has a much nicer answer if we instead try to interpret locales in the ground model as spaces in the extension, which can be thought of as generalized spaces where we remember only the structure of the open sets. For instance, in contrast to the setting of interpreting spaces as spaces, the product of locales always interprets to the product of spaces. Some nice corollaries are new forcing based proofs to certain locales being nonspatial. Many well-studied properties of locales transmit precisely to the corresponding notions of spaces: a locale is Hausdorff if and only if the corresponding space is Hausdorff in all forcing extensions and a locale is regular if and only if the corresponding space is $T_3$ in all forcing extensions, and more.