Category:Definitions/Product Topology

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This category contains definitions related to Product Topology.
Related results can be found in Category:Product Topology.

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 $\family {X_i}_{i \mathop \in I}$:

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

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

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

The product topology on $\XX$ is defined as the initial topology $\tau$ on $\XX$ with respect to $\family {\pr_i}_{i \mathop \in I}$.

That is, $\tau$ is the topology generated by:

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

where $\pr_i^{-1} \sqbrk U$ denotes the preimage of $U$ under $\pr_i$.