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

Proof
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 $\tau$.

For all $j \in I$ let:
 * $Z_j = \begin{cases} X_i & i = j \\

\set{z_j} & j \ne i \end{cases}$

Lemma 1
From Product Space of Subspaces is Subspace of Product Space, $\struct {Y_i, \upsilon_i}$ is a product space.

Consider the projection:
 * $p_i: \struct {Y_i, \upsilon_i} \to \struct {X_i, \tau_i}$

From Projection from Product of Family is Injection iff Other Factors are Singletons, $p_i$ is injective.

From Projection from Product of Family is Surjective, $p_i$ is surjective. From Projection from Product Topology is Continuous:General Result, $p_i$ is continuous.

From Projection from Product Topology is Open:General Result, $p_i$ is open.

Thus, by definition, we have that $p_i$ is a homeomorphism.

Consider the projection:
 * $\pr_i: \struct {X, \tau} \to \struct {X_i, \tau_i}$

and the restriction:
 * $\pr_i {\restriction_{Y_i} }: \struct {Y_i, \upsilon_i} \to \struct {X_i, \tau_i}$

Lemma 2
Thus $\pr_i {\restriction_{Y_i} }: Y_i \to X_i$ is a homeomorphism.