- #1
littleHilbert
- 56
- 0
Hi! I'd like to ask the following question.
Does it make sense to take unions and intersections over an empty set?
For instance I came across a definition of a topological space which uses just two axioms:
A topology on a set X is a subset T of the power set of X, which satisfies:
1. The union of any familiy of sets in T belongs to T. Applying this to the empty family, we obtain in particular [tex]\emptyset\in{}T[/tex]
2. The intersection of any finite family of sets in T belongs to T. Applying this to the empty family, we obtain in particular [tex]X\in{}T[/tex]
The empty family is just a family of sets with an empty index set, isn't it? Or did I misunderstand the notion of the empty family.
Does it make sense to take unions and intersections over an empty set?
For instance I came across a definition of a topological space which uses just two axioms:
A topology on a set X is a subset T of the power set of X, which satisfies:
1. The union of any familiy of sets in T belongs to T. Applying this to the empty family, we obtain in particular [tex]\emptyset\in{}T[/tex]
2. The intersection of any finite family of sets in T belongs to T. Applying this to the empty family, we obtain in particular [tex]X\in{}T[/tex]
The empty family is just a family of sets with an empty index set, isn't it? Or did I misunderstand the notion of the empty family.