Projection from Product Topology is Continuous/General Result

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Theorem

Let $\family {T_i}_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle T = \prod_{i \mathop \in I} T_i$ be the corresponding product space.

Let $\pr_i : T \to T_i$ be the corresponding projection from $T$ onto $T_i$.


Then $\pr_i$ is continuous for all $i \in I$.


Proof

Let $i \in I$.

Let $U \subseteq T_i$ be an open set.

Then by the definition of the topology of a product space, $\map {\pr_i^{-1} } U$ is an open set in $T$.

Thus $\pr_i$ is continuous.

$\blacksquare$