Product Topology is Coarsest Topology such that Projections are Continuous
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Theorem
Let $\mathbb X = \family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $X$ be the cartesian product of $\mathbb X$:
- $\ds X := \prod_{i \mathop \in I} X_i$
Let $\tau$ be the product topology on $X$.
For each $i \in I$, let $\pr_i : X \to X_i$ be the corresponding projection which maps each ordered tuple in $X$ to the corresponding element in $X_i$:
- $\forall x \in X: \map {\pr_i} x = x_i$
Then $\tau$ is the coarsest topology on $X$ such that all the $\pr_i$ are continuous.
Proof
The result follows from the definition of the product topology and Equivalence of Definitions of Initial Topology.
$\blacksquare$
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