The image of a linear operator has this property if and only if the operator’s adjoint is injective. The subspace spanned by finite linear combinations of a Schauder basis has this property. For a countable collection of sets that all have this property nowhere, their union is called “meager,” or first category. If open subsets of a complete space have this property, then so does their countable intersection, by the (*) Baire category theorem. A space has this property in any of its compactifications. A countable collection of polynomials has this property in C[0, 1] (“C zero one”) by the Stone-Weierstrass theorem, so the latter space is separable. Two continuous functions that agree on a set with this property must be the same, since adjoining limit points to one of these sets gives the whole space. For 10 points, give this property of a subset whose closure is the entire space. ■END■ (0[3])