Editing Product topology (section)

==Properties==

The set of Cartesian products between the open sets of the topologies of each <math>X_i</math> forms a basis for what is called the [[box topology]] on <math>X.</math> In general, the box topology is [[Finer topology|finer]] than the product topology, but for finite products they coincide.

The product space <math>X,</math> together with the canonical projections, can be characterized by the following [[universal property]]: if <math>Y</math> is a topological space, and for every <math>i \in I,</math> <math>f_i : Y \to X_i</math> is a continuous map, then there exists {{em|precisely one}} continuous map <math>f : Y \to X</math> such that for each <math>i \in I</math> the following diagram [[Commutative diagram|commutes]]:

{{bi|left=1.3em|[[File:CategoricalProduct-02.svg|frameless|upright=0.6|Characteristic property of product spaces|class=skin-invert]] }}

This shows that the product space is a [[Product (category theory)|product]] in the [[category of topological spaces]]. It follows from the above universal property that a map <math>f : Y \to X</math> is continuous [[if and only if]] <math>f_i = p_i \circ f</math> is continuous for all <math>i \in I.</math> In many cases it is easier to check that the component functions <math>f_i</math> are continuous. Checking whether a map <math> X \to Y</math> is continuous is usually more difficult; one tries to use the fact that the <math>p_i</math> are continuous in some way.

In addition to being continuous, the canonical projections <math>p_i : X \to X_i</math> are [[open map]]s. This means that any open subset of the product space remains open when projected down to the <math>X_i.</math> The converse is not true: if <math>W</math> is a [[Subspace (topology)|subspace]] of the product space whose projections down to all the <math>X_i</math> are open, then <math>W</math> need not be open in <math>X</math> (consider for instance <math display="inline">W = \R^2 \setminus (0, 1)^2.</math>) The canonical projections are not generally [[closed map]]s (consider for example the closed set <math display="inline">\left\{(x,y) \in \R^2 : xy = 1\right\},</math> whose projections onto both axes are <math>\R \setminus \{0\}</math>).

Suppose <math display="inline">\prod_{i \in I} S_i</math> is a product of arbitrary subsets, where <math>S_i \subseteq X_i</math> for every <math>i \in I.</math> If all <math>S_i</math> are {{em|non-empty}}  then <math display="inline">\prod_{i \in I} S_i</math> is a closed subset of the product space <math>X</math> if and only if every <math>S_i</math> is a closed subset of <math>X_i.</math> More generally, the closure of the product <math display="inline">\prod_{i \in I} S_i</math> of arbitrary subsets in the product space <math>X</math> is equal to the product of the closures:{{sfn|Bourbaki|1989|pp=43-50}}

<math display=block>
{\operatorname{Cl}_X}\Bigl(\prod_{i \in I} S_i\Bigr) = \prod_{i \in I} \bigl({\operatorname{Cl}_{X_i}} S_i\bigr).
</math>

Any product of [[Hausdorff space]]s is again a Hausdorff space.

[[Tychonoff's theorem]], which is equivalent to the [[axiom of choice]], states that any product of [[compact space]]s is a compact space. A specialization of [[Tychonoff's theorem]] that requires only [[the ultrafilter lemma]] (and not the full strength of the axiom of choice) states that any product of compact [[Hausdorff space|Hausdorff]] spaces is a compact space.

If <math display="inline">z = \left(z_i\right)_{i \in I} \in X</math> is fixed then the set

<math display=block>
\left\{ x = \left(x_i\right)_{i \in I} \in X \mathbin{\big\vert} x_i = z_i \text{ for all but finitely many } i \right\}
</math>

is a [[Dense set|dense subset]] of the product space <math>X</math>.{{sfn|Bourbaki|1989|pp=43-50}}