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I-DENSITY CONTINUOUS FUNCTIONS Krzysztof Ciesielski他 1994- 被引用数: 84
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CHAPTER 1
The Ordinary Density Topology
1.1. A Simple Category Topology
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.
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