To gain some insight into what is happening with limits like this, it is useful to generalize this idea to a topological setting. A nonempty family J ⊂P(X) of subsets of X is an ideal on X if A ⊂ B and B ∈ J imply that A ∈ J and if A∪B ∈ J provided A,B ∈ J. An ideal J on X is said to be a σ-ideal on X if ∪n∈N An ∈ J for every family {An : n ∈ N} ⊂ J. Let J be an ideal on R and Ｔo be the ordinary topology on R. The set T (J) = {G ＼ J : G ∈ Ｔo, J ∈ J} is a topology on R which is finer than Ｔo. The following proposition is evident from the definitions.
Proposition 1.1.1. Let J be a σ-ideal on R and T (J ) be as above. For f : (R, T (J )) → (R, Ｔo) and x0 ∈ R the following statements are equivalent to each other. (i): f is continuous at x0. (ii): Given ε > 0 there is a δ > 0 such that {x ∈ (x0 − δ, x0 + δ) : |f(x) − f(x0)| ≧ ε} ∈ J.
