Category Argument Template
A standard Baire category method for producing a dense or residual set by intersecting dense open sets.
Category argument template. Let be a Baire space and let be open and dense. Then
is dense in , and it is a residual set because its complement is a countable union of nowhere dense sets. If , the intersection is therefore nonempty.
How to use the method
Conceptually, this packages the Baire category theorem into a reusable method: encode the th requirement of a property as membership in a dense open set , and then conclude that there are points satisfying all requirements at once (finite versions rely only on intersection of dense open sets being dense).