The standard topology on ℝ is generated by the collection of all open intervals (a, b) with a < b. Which of the following is an equally valid basis for the same topology?
AAll closed intervals [a, b] with a < b
BAll open intervals (a, b) with a and b rational
CAll half-open intervals [a, b) with a < b
DAll open intervals (a, b) with b − a > 1
The key test: does each basis element of the new collection lie in some union of elements from the original, and vice versa? Every open interval (a, b) with rational endpoints is itself an open interval (a subset of all open intervals), and every open interval with real endpoints can be written as a union of open intervals with rational endpoints (by density of rationals). So the rational-endpoint intervals generate the same topology. Closed intervals fail because they are not open sets in the standard topology. Half-open intervals generate the lower-limit (Sorgenfrey) topology, a strictly finer topology.
Question 2 Multiple Choice
A collection B satisfies both basis axioms on a set X. What is the topology generated by B?
AB itself, since B already satisfies the topology axioms
BAll finite intersections of elements of B, plus the empty set
CAll arbitrary unions of elements of B, plus the empty set
DAll elements of B together with their complements
The topology generated by a basis B consists of all arbitrary unions of basis elements (plus the empty set, which is the empty union). The basis axioms — every point is in some basis element, and intersections of basis elements contain a basis element around each point — are precisely what guarantee that taking all unions yields a collection satisfying the topology axioms. Note that B itself need not be closed under unions, so B alone is generally not the topology.
Question 3 True / False
The standard topology on ℝ can be generated by a countable basis — for example, the collection of all open intervals with rational endpoints.
TTrue
FFalse
Answer: True
This is correct and important. The collection {(p, q) : p, q ∈ ℚ, p < q} is countable (a countable product of countable sets) and generates the standard topology on ℝ. Every open interval (a, b) with real endpoints is a union of rational-endpoint intervals (by density of rationals in ℝ). A space that admits a countable basis is called second countable, a key property used throughout analysis and topology.
Question 4 True / False
A basis for a topology is expected to itself be a topology — that is, it should be closed under arbitrary unions and finite intersections.
TTrue
FFalse
Answer: False
A basis does not need to be a topology. A basis B is required to satisfy two simpler conditions: (1) every point lies in some basis element, and (2) the intersection of any two basis elements contains a basis element around each point in the intersection. Condition (2) is weaker than requiring B to be closed under intersections — B₁ ∩ B₂ itself need not be in B, only some smaller B₃ must be. The topology generated by B is larger than B itself: it consists of all unions of basis elements.
Question 5 Short Answer
What does it mean for two bases B and B' to generate the same topology, and how would you verify this?
Think about your answer, then reveal below.
Model answer: Two bases B and B' generate the same topology if and only if every element of B can be expressed as a union of elements of B', and conversely every element of B' can be expressed as a union of elements of B. In practice: for each B ∈ B and each point x ∈ B, there must exist some B' ∈ B' with x ∈ B' ⊆ B; and symmetrically for each B' ∈ B' and each point x ∈ B'. This mutual containment condition ensures that the set of all unions of B-elements equals the set of all unions of B'-elements.
This criterion is far more tractable than comparing the full topologies directly (which would involve verifying equality of two potentially uncountable collections). The check is local: for each basis element and each point in it, find a smaller basis element from the other family. If this works in both directions, the families generate the same open sets. For example, all open intervals and all rational-endpoint intervals both pass this check against each other, confirming they generate the standard topology on ℝ.