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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]].
[[Image:CategoricalProduct-02.png|center|Characteristic property of product spaces]]
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>
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>W = \mathbb{R}^2 \setminus (0, 1)^2.</math>) The canonical projections are not generally [[closed map]]s (consider for example the closed set <math>\left\{ (x,y) \in \mathbb{R}^2 : xy = 1 \right\},</math> whose projections onto both axes are <math>\mathbb{R} \setminus \{ 0 \}</math>).
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