
“real: chapter_04” — 2011/5/22 — 23:17 — page 40 — #40
4-40 Real Analysis
A collection τ of subsets of X can now be defined as follows.
B ∈ τ if and only if either B is empty or for each x ∈ B there exists
U ∈
C such that x ∈ U ⊂ B. Then (X , τ) is a topological space. The
collection
C will also be called a basis for τ .
Proof The required axioms can be easily verified.
Note 4.3.54 Consider the extended real number system [−∞, ∞] =
R ∪ {−∞, +∞}. Let C be the collection of all intervals of the form
(a, b) with −∞ ≤ a < b ≤∞together with the sets of the form
[−∞, a) = (−∞, a) ∪ {−∞} (a ∈
R) and (b, ∞] = (b, ∞) ∪ {+∞}
(b ∈
R). Obviously, C is a collection