A function f: Z → X × Y is given. Which condition is both necessary and sufficient for f to be continuous in the product topology?
Af maps every open set in Z to an open set in X × Y
BBoth component functions π₁ ∘ f: Z → X and π₂ ∘ f: Z → Y are continuous
Cf is injective and preserves the metric on X × Y
Df maps basis elements of Z to basis elements of X × Y
The product topology is defined as the coarsest topology making the projections continuous — and this characterization passes directly to maps into the product: f is continuous if and only if each component is continuous. This is the universal property of the product topology. Option A describes an open map, not a continuous one. Option C imposes conditions (injectivity, metric) that have nothing to do with continuity. Option D is not a general continuity criterion.
Question 2 Multiple Choice
For an infinite product ∏ᵢ Xᵢ, the box topology differs from the product topology by allowing open sets where *every* coordinate is restricted (Uᵢ ≠ Xᵢ for all i). Which consequence does this difference produce?
AThe box topology makes projections discontinuous, so it fails to be a valid topology
BThe box topology is strictly finer than the product topology and fails to preserve compactness — Tychonoff's theorem does not hold for it
CThe box topology is strictly coarser than the product topology, so it has weaker separation properties
DThe box topology and product topology agree on all separability and compactness properties but differ only on convergence
The box topology is strictly *finer* (more open sets) than the product topology, not coarser. Having more open sets means open covers are easier to find but harder to reduce to finite subcovers — Tychonoff's theorem fails. The product topology's constraint that only finitely many coordinates are restricted is precisely what forces compactness to be preserved. Coarser topologies have stronger convergence properties; the product topology's 'smallness' is a feature, not a limitation.
Question 3 True / False
In the product topology on ℝ × ℝ, the set (0, 1) × (0, 1) is an open set.
TTrue
FFalse
Answer: True
Open rectangles U × V, where U is open in X and V is open in Y, form a *basis* for the product topology — they are the building blocks from which all open sets are constructed. (0,1) × (0,1) is exactly such a basis element: (0,1) is open in ℝ and so is (0,1). The product topology on ℝ × ℝ coincides with the standard Euclidean topology on ℝ², in which open rectangles are indeed open.
Question 4 True / False
The box topology on an infinite product has strictly fewer open sets than the product topology.
TTrue
FFalse
Answer: False
This is backwards. The box topology is strictly *finer* — it has strictly *more* open sets. In the product topology, a basis element πᵢ⁻¹(Uᵢ) restricts only one coordinate and leaves all others as the full space; finite intersections allow finitely many restricted coordinates. The box topology allows *infinitely many* restricted coordinates simultaneously, generating many more open sets. The product topology is the *coarsest* topology making projections continuous; the box topology is strictly coarser.
Question 5 Short Answer
The product topology is defined as the *coarsest* topology making projections continuous. Why does 'coarsest' matter, and what fails if you use a finer topology instead?
Think about your answer, then reveal below.
Model answer: The coarsest topology has the fewest open sets consistent with the requirement that projections are continuous. Using a finer topology (like the box topology for infinite products) adds more open sets than the continuity requirement demands. This has consequences: in the product topology, a sequence converges if and only if each coordinate converges — a clean, useful characterization. In the box topology, this fails. More importantly, Tychonoff's theorem (a product of compact spaces is compact) holds only for the product topology; the box topology breaks compactness. Coarser topologies enforce stronger convergence and compactness properties by making it harder to separate points.
The 'coarsest topology' criterion is not an arbitrary choice — it is the unique topology satisfying a universal property: every continuous map into the product factors through the projections. This universality is what makes the product topology 'right.' Finer topologies satisfy continuity of projections but impose additional open sets that are not forced by the requirement, losing the structural benefits that make the product topology so useful in analysis and topology.