Definition:Product Topology/Natural Sub-Basis

Definition

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.

Let $\XX$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:

$\ds \XX := \prod_{i \mathop \in I} X_i$

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For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$:

$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$

The natural sub-basis on $\XX$ is defined as:

$\SS = \set {\pr_i^{-1} \sqbrk U: i \in I, \, U \in \tau_i}$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions