Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Injection

Theorem

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

Let $\ds X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$.

Let $z \in X$.

Let $i \in I$.

Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.

Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$.

Then:

$p_i$ is an injection.

Proof

Note that by definitions of a restriction and a projection then:

$\forall y \in Y_i: \map {p_i} y = y_i$

Let $x, y \in Y_i$.

Then for all $j \in I \setminus \set i$:

$x_j = z_j = y_j$

Let $\map {p_i} x = \map {p_i} y$.

Then:

$x_i = y_i$

Thus:

$x = y$

It follows that $p_i$ is an injection by definition.

$\blacksquare$