You want to define a topology on X × Y such that the projection maps πX: X × Y → X and πY: X × Y → Y are both continuous. What is the natural subbasis for this topology?
AAll subsets of X × Y
BSets of the form U × V where U is open in X and V is open in Y
CSets of the form U × Y where U is open in X, together with sets of the form X × V where V is open in Y
DAll open sets in either X or Y, treated directly as subsets of X × Y
For πX to be continuous, every open U in X must have πX⁻¹(U) = U × Y open in the product. For πY to be continuous, every open V in Y must have πY⁻¹(V) = X × V open. These cylinder sets form the natural subbasis. Their finite intersections (U × Y) ∩ (X × V) = U × V form the standard product basis. Option B gives the basis directly — but the subbasis is the cleaner conceptual description of what continuity of projections requires.
Question 2 Multiple Choice
A subbasis S for a topology on X is:
AA collection of sets whose arbitrary unions give all open sets directly
BA collection of sets whose finite intersections form a basis, from which the topology is generated by taking arbitrary unions
CAny collection of subsets of X, with no conditions required
DA collection of sets that are simultaneously open and closed in the topology
The subbasis requires a two-step construction: finite intersections of subbasis elements form a basis, and then arbitrary unions of basis elements give the full topology. Option A describes a basis, not a subbasis. Option C is almost right — subbases do have minimal conditions (typically that they cover X) — but the key defining feature is the two-step generation process.
Question 3 True / False
Every basis for a topology on X is also a subbasis for that topology, but not every subbasis is a basis.
TTrue
FFalse
Answer: True
A basis already satisfies the conditions needed to generate the topology by unions. Since finite intersections of basis elements can be covered by basis elements (the basis axiom), a basis also satisfies the subbasis condition. But a subbasis need not be closed under finite intersections in the basis sense — its elements might not cover intersections with basis elements — so a subbasis is the strictly more general concept.
Question 4 True / False
The topology generated by a subbasis S consists exactly of most finite intersections of elements of S, together with ∅ and X.
TTrue
FFalse
Answer: False
Finite intersections of subbasis elements give the basis, not the full topology. The topology also includes all arbitrary unions of those finite intersections. For example, U₁ ∩ U₂ and V₁ ∩ V₂ (subbasis intersections) are basis elements, but their union (U₁ ∩ U₂) ∪ (V₁ ∩ V₂) is open in the topology without being a subbasis intersection. The full construction is: subbasis → finite intersections → basis → arbitrary unions → topology.
Question 5 Short Answer
Why is the subbasis — rather than directly specifying a basis — the natural tool for defining the product topology or initial topologies?
Think about your answer, then reveal below.
Model answer: The subbasis lets you declare which sets you want to be open (cylinder sets for the product; preimages of open sets for initial topologies) without immediately requiring closure under finite intersections. This is cleaner conceptually: you state what openness means intuitively, and the two-step construction handles the closure requirements. For the product topology, cylinder sets U × Y and X × V naturally arise from continuity of projections; their intersections U × V (the basis) follow automatically.
The subbasis is the correct primitive for initial topologies because you are constructing the coarsest topology making a family of maps continuous. You pull back open sets from each target space and declare those preimages to be open — that's the subbasis. You don't need to explicitly compute all intersections and unions; the subbasis construction does it for you. This is why subbases appear wherever topologies are defined by universal properties, including product, subspace, and quotient constructions.