Subspace of Product Space has Initial Topology with respect to Restricted Projections

Theorem

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 projection of $\XX$ onto $X_i$.

Let $Y \subseteq \XX$ be a non-empty subset of $\XX$.

For each $i \in I$, let $\pr_i \restriction_Y$ denote the restriction of $\pr_i$ to $Y$.

Let $\tau_\XX$ be the product topology on $\XX$.

Let $\tau_Y$ be the subspace topology on $Y$.

Then:

$\tau_Y$ is the initial topology on $Y$ with respect to the mappings $\family {\pr_i \restriction_Y : Y \to X_i}_{i \mathop \in I}$

Proof

By definition of product topology:

$\tau_\XX$ is the initial topology with respect to the mappings $\family {\pr_i : \XX \to X_i}_{i \mathop \in I}$

From Subspace Topology on Initial Topology is Initial Topology on Restrictions:

$\tau_Y$ is the initial topology on $Y$ with respect to the mappings $\family {\pr_i \restriction_Y : Y \to X_i}_{i \mathop \in I}$

$\blacksquare$