Subspace of Product Space is Homeomorphic to Factor Space/Proof 2/Lemma 1

Theorem
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set.

Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.

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 $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.

Let $p_i = \pr_i {\restriction_{Y_i}}$.

Let $J$ be a finite subset of $I$.

Then:
 * $\bigcap_{k \in J} \map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$

Proof
Let $x \in X_i$.

Let $k \in I_Y$.

Then:

Case 1: $k = i$
Let $k = i$.

Then:

So:
 * $x \in \map {p_i} {\map{\pr_i^\gets } {V_i} \cap Y_i} \iff x \in V_i$

That is:
 * $\map {p_i} {\map{\pr_i^\gets } {V_i} \cap Y_i} = V_i \in \tau_i$

Case 2: $k \neq i$
In either case:
 * $\map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$.

Since $k \in I_y$ was arbitrary, then:
 * $\forall k \in I_y: \map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$

By open set axiom $O2$, then:
 * $\bigcap_{k \in I_y} \map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$