Topology/Product Spaces

We briefly review the set-theoretic notion of a (possibly infinite) Cartesian product. Our formulation uses functions, which makes arbitrary index sets painless.

Let ${\displaystyle J}$ be an index set, and for each ${\displaystyle \alpha \in J}$ let ${\displaystyle X_{\alpha }}$ be a set. The (set-theoretic) Cartesian product

${\displaystyle \prod _{\alpha \in J}X_{\alpha }}$

is the set of all functions ${\displaystyle x:J\to \bigcup _{\alpha \in J}X_{\alpha }}$ with ${\displaystyle x(\alpha )\in X_{\alpha }}$ for every ${\displaystyle \alpha }$. We write such an element as ${\displaystyle x=(x_{\alpha })_{\alpha \in J}}$, where ${\displaystyle x_{\alpha }:=x(\alpha )}$.

• If all ${\displaystyle X_{\alpha }=X}$, then ${\displaystyle \prod _{\alpha \in J}X_{\alpha }=X^{J}}$, the set of all ${\displaystyle J}$-tuples in ${\displaystyle X}$.
• (Empty product.) If ${\displaystyle J=\varnothing }$, by convention ${\displaystyle \prod _{\alpha \in J}X_{\alpha }}$ is a singleton set. Topologically this will become the one-point space.

Let ${\displaystyle J=\mathbb {N} }$ and ${\displaystyle X_{\alpha }=\mathbb {R} }$ for each ${\displaystyle \alpha }$. Then

${\displaystyle \prod _{\alpha \in J}X_{\alpha }=\mathbb {R} ^{\mathbb {N} }=\{(x_{1},x_{2},\dotsc )\mid x_{n}\in \mathbb {R} \ \forall n\}}$.

Let ${\displaystyle (X_{\alpha },{\mathcal {T}}_{\alpha })}$ be topological spaces indexed by ${\displaystyle J}$. For each ${\displaystyle \beta \in J}$, define the projection

${\displaystyle \pi _{\beta }:\prod _{\alpha \in J}X_{\alpha }\to X_{\beta },\qquad \pi _{\beta }((x_{\alpha }))=x_{\beta }.}$

There are two natural topologies one can put on the product set.

The box topology on ${\displaystyle \prod _{\alpha \in J}X_{\alpha }}$ has as a basis all products

${\displaystyle \prod _{\alpha \in J}U_{\alpha }}$

with each ${\displaystyle U_{\alpha }}$ open in ${\displaystyle X_{\alpha }}$ (no finiteness restriction).

The product topology on ${\displaystyle \prod _{\alpha \in J}X_{\alpha }}$ is the topology initial with respect to the family of projections ${\displaystyle \{\pi _{\beta }\}_{\beta \in J}}$: it is the coarsest topology making every projection continuous.

Equivalently, it is generated by the subbasis

${\displaystyle {\mathcal {S}}=\{\ \pi _{\beta }^{-1}(U)\ :\ \beta \in J,\ U\subseteq X_{\beta }{\text{ open}}\ \}.}$

Taking finite intersections of subbasic sets yields the familiar basis description:

A basis for the product topology consists of all sets

${\displaystyle \prod _{\alpha \in J}U_{\alpha }}$

with each ${\displaystyle U_{\alpha }}$ open in ${\displaystyle X_{\alpha }}$ and ${\displaystyle U_{\alpha }=X_{\alpha }}$ for all but finitely many indices.

A basis for the box topology is the same formula without the finiteness condition.

Consequences. For finite products the two topologies coincide. In general, the box topology is finer (has more open sets) than the product topology.

If ${\displaystyle {\mathcal {B}}_{\alpha }}$ is a basis for ${\displaystyle X_{\alpha }}$, then:

• ${\displaystyle \{\prod _{\alpha }B_{\alpha }:B_{\alpha }\in {\mathcal {B}}_{\alpha }\ \forall \alpha \}}$ is a basis for the box topology.
• ${\displaystyle \{\prod _{\alpha }B_{\alpha }:B_{\alpha }\in {\mathcal {B}}_{\alpha }\ {\text{for finitely many }}\alpha ,\ B_{\alpha }=X_{\alpha }\ {\text{otherwise}}\}}$ is a basis for the product topology.

Many fundamental theorems extend from finite to arbitrary products only for the product topology. The key mechanism is the initial-topology/“universal property” viewpoint below, which forces continuity and limit behavior to be coordinatewise. The box topology is too fine for these extensions and is chiefly useful for counterexamples.

We now prove the central criterion for continuity into a product space and state its categorical formulation.

Let ${\displaystyle (X_{\alpha },{\mathcal {T}}_{\alpha })}$ be spaces indexed by ${\displaystyle J}$, give ${\displaystyle \prod _{\alpha \in J}X_{\alpha }}$ the product topology, and let ${\displaystyle Y}$ be a space. For a map ${\displaystyle f:Y\to \prod _{\alpha }X_{\alpha }}$, write ${\displaystyle f_{\beta }:=\pi _{\beta }\circ f:Y\to X_{\beta }}$. Then

${\displaystyle f}$ is continuous if and only if each ${\displaystyle f_{\beta }}$ is continuous.

Proof.

• “Only if”: Each projection ${\displaystyle \pi _{\beta }}$ is continuous (by definition of the product topology via the subbasis ${\displaystyle {\pi _{\beta }^{-1}(U)}}$). Hence ${\displaystyle f_{\beta }=\pi _{\beta }\circ f}$ is continuous as a composition.
• “If”: It suffices to show that the preimage under ${\displaystyle f}$ of every subbasic open set is open. A typical subbasic set is ${\displaystyle \pi _{\beta }^{-1}(U)}$ with ${\displaystyle U\subseteq X_{\beta }}$ open. Then

${\displaystyle f^{-1}\!{\big (}\pi _{\beta }^{-1}(U){\big )}=(\pi _{\beta }\circ f)^{-1}(U)=f_{\beta }^{-1}(U),}$

which is open by continuity of ${\displaystyle f_{\beta }}$. Since finite intersections of such sets form a basis, ${\displaystyle f}$ is continuous. ∎

Given a family of maps ${\displaystyle (g_{\alpha }:Y\to X_{\alpha })_{\alpha \in J}}$, there is a unique function

${\displaystyle g:Y\to \prod _{\alpha }X_{\alpha },\qquad g(y)=(g_{\alpha }(y))_{\alpha \in J},}$

and ${\displaystyle g}$ is continuous if and only if each ${\displaystyle g_{\alpha }}$ is continuous.

Remark. The forward direction (“continuous into the product ⇒ all coordinates continuous”) holds for the box topology as well. The reverse fails in general for infinite products with the box topology when the product is infinite. For example, the diagonal map ${\displaystyle f:\mathbb {R} \to \mathbb {R} ^{\mathbb {N} }}$, ${\displaystyle f(t)=(t,t,t,\dots )}$ has continuous coordinates, yet is not continuous for the box topology (preimage of ${\displaystyle (-1,1)\times (-{\tfrac {1}{2}},{\tfrac {1}{2}})\times \cdots }$ would have to contain an interval around ${\displaystyle 0}$, which it does not).

Let ${\displaystyle D:J\to \mathbf {Top} }$ be the diagram selecting the family ${\displaystyle (X_{\alpha })}$. A cone to ${\displaystyle D}$ with vertex ${\displaystyle Y}$ is a family of morphisms ${\displaystyle (f_{\alpha }:Y\to X_{\alpha })}$. The product object is characterized up to unique isomorphism by the following:

Proposition. With the product topology on ${\displaystyle \prod _{\alpha }X_{\alpha }}$ and projections ${\displaystyle \pi _{\alpha }}$, the pair ${\displaystyle {\big (}\prod _{\alpha }X_{\alpha },(\pi _{\alpha }){\big )}}$ is terminal in the category of cones over ${\displaystyle D}$: for every cone ${\displaystyle (Y,(f_{\alpha }))}$ there exists a unique continuous map

${\displaystyle u:Y\to \prod _{\alpha }X_{\alpha }}$

such that ${\displaystyle \pi _{\alpha }\circ u=f_{\alpha }}$ for all ${\displaystyle \alpha }$.

Proof.

• Existence/uniqueness as a set map: Define ${\displaystyle u(y)=(f_{\alpha }(y))_{\alpha }}$; this is the only function with the required coordinate identities, since equality in the product is coordinatewise.
• Continuity: By the theorem above, ${\displaystyle u}$ is continuous if and only if each ${\displaystyle f_{\alpha }}$ is continuous; but morphisms in ${\displaystyle \mathbf {Top} }$ are precisely continuous maps, so ${\displaystyle u}$ is a valid arrow in ${\displaystyle \mathbf {Top} }$.

Thus ${\displaystyle (\prod X_{\alpha },\pi _{\alpha })}$ is terminal among cones. ∎

Equivalently, the product topology is the initial topology on the set ${\displaystyle \prod _{\alpha }X_{\alpha }}$ with respect to the family of projections ${\displaystyle (\pi _{\alpha })}$: it is the coarsest topology making all ${\displaystyle \pi _{\alpha }}$ continuous. The continuity criterion above is exactly the defining property of an initial topology.

• Projections are continuous and open. Each ${\displaystyle \pi _{\beta }}$ is continuous (by construction) and maps a basic open box ${\displaystyle \prod U_{\alpha }}$ to ${\displaystyle U_{\beta }}$, hence is an open map. Projections need not be closed in general; they are closed if all other factors are compact and the target is Hausdorff (see the compactness section for context).
• Subspaces. For subsets ${\displaystyle A_{\alpha }\subseteq X_{\alpha }}$, the canonical identification

${\displaystyle \prod _{\alpha }A_{\alpha }\ \subseteq \ \prod _{\alpha }X_{\alpha }}$

is a subspace embedding for both box and product topologies.

• Separation. If every ${\displaystyle X_{\alpha }}$ is ${\displaystyle T_{1}}$ (resp. Hausdorff), then so is ${\displaystyle \prod _{\alpha }X_{\alpha }}$ (for either topology). (Normality does not pass to products in general; e.g. the Sorgenfrey plane.)
• Connectedness / path-connectedness. Products of connected spaces are connected; products of path-connected spaces are path-connected (choose coordinatewise paths and use the universal property).
• Convergence. In the product topology, a sequence (or net) ${\displaystyle (x^{(n)})}$ converges to ${\displaystyle x}$ iff each coordinate sequence (or net) ${\displaystyle (\pi _{\alpha }(x^{(n)}))}$ converges to ${\displaystyle \pi _{\alpha }(x)}$. (This coordinatewise description fails for the box topology.)
• Closures. For either topology one has

${\displaystyle {\overline {\prod _{\alpha }A_{\alpha }}}=\prod _{\alpha }{\overline {A_{\alpha }}}.}$

We give prove of the last property above, that closures commute with products in both the product and the box topology (the rest are left as an exercise).

Theorem. Let ${\displaystyle (X_{\alpha }){\alpha \in J}}$ be spaces and ${\displaystyle A\alpha \subseteq X_{\alpha }}$. If ${\displaystyle \prod X_{\alpha }}$ is given either the product topology or the box topology, then

${\displaystyle {\overline {\prod _{\alpha \in J}A_{\alpha }}}\;=\;\prod _{\alpha \in J}{\overline {A_{\alpha }}}.}$

Proof. We show the two inclusions.

(⊆) Let ${\displaystyle x=(x_{\alpha })\in \prod _{\alpha }{\overline {A_{\alpha }}}}$. Let ${\displaystyle U}$ be a basic open neighborhood of ${\displaystyle x}$. For the product topology, we may write ${\displaystyle U=\prod _{\alpha }U_{\alpha }}$ with each ${\displaystyle U_{\alpha }}$ open in ${\displaystyle X_{\alpha }}$ and ${\displaystyle U_{\alpha }=X_{\alpha }}$ for all but finitely many indices; for the box topology, we have the same description but without the finiteness restriction.

Since ${\displaystyle x_{\alpha }\in {\overline {A_{\alpha }}}}$ and ${\displaystyle U_{\alpha }}$ is an open neighborhood of ${\displaystyle x_{\alpha }}$, we have ${\displaystyle U_{\alpha }\cap A_{\alpha }\neq \varnothing }$ for every ${\displaystyle \alpha }$. Choose ${\displaystyle y_{\alpha }\in U_{\alpha }\cap A_{\alpha }}$ for each ${\displaystyle \alpha }$, and set ${\displaystyle y=(y_{\alpha })}$. Then ${\displaystyle y\in U\cap \prod _{\alpha }A_{\alpha }}$, so every basic neighborhood of ${\displaystyle x}$ meets ${\displaystyle \prod _{\alpha }A_{\alpha }}$. Hence ${\displaystyle x\in {\overline {\prod _{\alpha }A_{\alpha }}}}$.

(⊇) Let ${\displaystyle x\in {\overline {\prod _{\alpha }A_{\alpha }}}}$. Fix ${\displaystyle \beta \in J}$ and an open set ${\displaystyle V_{\beta }\subseteq X_{\beta }}$ with ${\displaystyle x_{\beta }\in V_{\beta }}$. The set

${\displaystyle W:=\pi _{\beta }^{-1}(V_{\beta })}$

is open in either topology and contains ${\displaystyle x}$; by assumption, ${\displaystyle W\cap \prod _{\alpha }A_{\alpha }\neq \varnothing }$. Pick ${\displaystyle z=(z_{\alpha })\in W\cap \prod _{\alpha }A_{\alpha }}$. Then ${\displaystyle z_{\beta }\in V_{\beta }\cap A_{\beta }}$, so ${\displaystyle V_{\beta }\cap A_{\beta }\neq \varnothing }$. Since ${\displaystyle V_{\beta }}$ was arbitrary, ${\displaystyle x_{\beta }\in {\overline {A_{\beta }}}}$. As this holds for each ${\displaystyle \beta }$, we conclude ${\displaystyle x\in \prod _{\alpha }{\overline {A_{\alpha }}}}$. ∎

Remark (on choice). In the first inclusion we selected one point ${\displaystyle y_{\alpha }\in U_{\alpha }\cap A_{\alpha }}$ for each index ${\displaystyle \alpha }$. For infinite index sets this uses the Axiom of Choice (indeed, “every product of nonempty sets is nonempty” is equivalent to Choice). No choice is needed for finite products, and the reverse inclusion does not require Choice.

Corollary (density passes to products). If each ${\displaystyle A_{\alpha }}$ is dense in ${\displaystyle X_{\alpha }}$, then ${\displaystyle \prod _{\alpha }A_{\alpha }}$ is dense in ${\displaystyle \prod _{\alpha }X_{\alpha }}$ in both the product and the box topology.

Often each factor has a convenient basis. For Euclidean space,

• A basis of ${\displaystyle \mathbb {R} }$ is the family of open intervals; thus a basis for ${\displaystyle \mathbb {R} ^{n}}$ (with the product topology) is all rectangles ${\displaystyle (a_{1},b_{1})\times \cdots \times (a_{n},b_{n})}$.
• For a countable product ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$, a basic open set is a “cylinder” where only finitely many coordinates are restricted to intervals, the rest are all of ${\displaystyle \mathbb {R} }$.

• Let ${\displaystyle J=\{1,2\}}$ and ${\displaystyle X_{\alpha }=\mathbb {R} }$ with the usual topology. Then the basic open sets of ${\displaystyle \mathbb {R} ^{2}}$ are rectangles ${\displaystyle (a,b)\times (c,d)}$:

• Let ${\displaystyle J=\{1,2\}}$ and ${\displaystyle X_{\alpha }=R_{\ell }}$ (the Sorgenfrey line). Then a basis for the product (the “Sorgenfrey plane”) has sets of the form ${\displaystyle [a,b)\times [c,d)}$:

It is helpful to frame products using category theory.

• Top as a category. Objects are spaces; morphisms are continuous maps.

• Categorical product. An object ${\displaystyle P}$ with morphisms ${\displaystyle p_{\beta }:P\to X_{\beta }}$ is a product of the family ${\displaystyle (X_{\beta })}$ if for every space ${\displaystyle Y}$ and maps ${\displaystyle f_{\beta }:Y\to X_{\beta }}$ there is a unique ${\displaystyle f:Y\to P}$ with ${\displaystyle p_{\beta }\circ f=f_{\beta }}$ for all ${\displaystyle \beta }$.

In Top, the set-theoretic product with the product topology and its projections satisfies this universal property. The box topology does not (for infinite families), precisely because the “⇐” direction in the continuity theorem can fail.

• Initial topologies. Given a family of maps ${\displaystyle (g_{i}:Y\to Z_{i})}$, the initial topology on ${\displaystyle Y}$ makes all ${\displaystyle g_{i}}$ continuous and is the coarsest such topology. The product topology is the initial topology of the projections ${\displaystyle (\pi _{\beta })}$. Dually, quotient topologies are final topologies.
• Functoriality. Products are functorial: a family of maps ${\displaystyle (h_{\alpha }:X_{\alpha }\to Y_{\alpha })}$ induces

${\displaystyle \prod _{\alpha }h_{\alpha }:\ \prod _{\alpha }X_{\alpha }\longrightarrow \prod _{\alpha }Y_{\alpha },\qquad (x_{\alpha })\mapsto (h_{\alpha }(x_{\alpha })),}$

which is continuous by the coordinatewise criterion.

• Limits. Top has all small limits. In practice, limits are computed as subspaces of products cut out by equalizer conditions. This viewpoint systematically explains many “product + equations” constructions.
• Exponentials. Unlike Set or CGHaus, the category Top is not cartesian closed in general: a right adjoint to ${\displaystyle (-)\times X}$ (an “exponential” ${\displaystyle Y^{X}}$) need not exist. In well-behaved subcategories (e.g. compactly generated Hausdorff spaces), it does.

1. Show that the families described under “Bases from bases” are indeed bases.
2. Verify that ${\displaystyle \prod _{\alpha }A_{\alpha }}$ with the subspace topology from ${\displaystyle \prod _{\alpha }X_{\alpha }}$ equals the product of the subspaces ${\displaystyle A_{\alpha }}$.
3. Prove the Hausdorff product theorem.
4. Show ${\displaystyle (X_{1}\times \cdots \times X_{n-1})\times X_{n}\cong X_{1}\times \cdots \times X_{n}}$.
5. Find which direction of the coordinatewise-continuity theorem remains true for the box topology (hint: it’s the easy one).
6. For ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ with the product topology, prove that sequence convergence is coordinatewise. Does your proof still work for the box topology?
7. Let ${\displaystyle \mathbb {R} _{\infty }\subset \mathbb {R} ^{\mathbb {N} }}$ be the sequences that are eventually zero. Compute its closure in the product topology and in the box topology.
8. For sequences ${\displaystyle (a_{i})}$ with ${\displaystyle a_{i}>0}$ and ${\displaystyle (b_{i})}$ real, show that ${\displaystyle h((x_{i}))=(a_{i}x_{i}+b_{i})}$ is a homeomorphism of ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$ in the product topology. What fails in the box topology?
9. (Universal property practice) Given a family ${\displaystyle f_{\alpha }:A\to X_{\alpha }}$, show there is a unique continuous ${\displaystyle f:A\to \prod _{\alpha }X_{\alpha }}$ with ${\displaystyle \pi _{\alpha }\circ f=f_{\alpha }}$, and identify the subspace topology on ${\displaystyle f(A)}$ that makes this universal for maps from a space ${\displaystyle Y}$ into ${\displaystyle A}$.