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form a [[subbase]] for the topology on <math>X.</math> A [[subset]] of <math>X</math> is open if and only if it is a (possibly infinite) [[Union (set theory)|union]] of [[Intersection (set theory)|intersections]] of finitely many sets of the form <math>p_i^{-1}\left(U_i\right).</math> The <math>p_i^{-1}\left(U_i\right)</math> are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.
The product topology is also called the {{em|[[topology of pointwise convergence]]}} because
Explicitly, a sequence <math display="inline">s_{\bull} = \left(s_n\right)_{n=1}^{\infty}</math> (respectively, a net <math display="inline">s_{\bull} = \left(s_a\right)_{a \in A}</math>) converges to a given point <math display="inline">x \in \prod_{i \in I} X_i</math> if and only if <math>p_i\left(s_{\bull}\right) \to p_i(x)</math> in <math>X_i</math> for every index <math>i \in I,</math> where <math>p_i\left(s_{\bull}\right) := p_i \circ s_{\bull}</math> denotes <math>\left(p_i\left(s_n\right)\right)_{n=1}^{\infty}</math> (respectively, denotes <math>\left(p_i\left(s_a\right)\right)_{a \in A}</math>).
In particular, if <math>X_i = \R</math> is used for all <math>i</math> then the Cartesian product is the space <math display="inline">\prod_{i \in I} \R = \R^I</math> of all [[Real number|real]]-valued [[Function (mathematics)|function]]s on <math>I,</math> and convergence in the product topology is the same as [[Pointwise convergence|pointwise convergence]] of functions.
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