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

Theorem

 * $\pr_i {\restriction_{Y_i} } = \pr'_i$

Proof
Let $z \in X$.

Let $i \in i$.

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

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

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

Lemma 1

 * $Y_i = \prod_{j \mathop \in I} Z_j$

Consider the projection:
 * $\pr'_i: Y_i \to X_i$

From Leigh.Samphier/Sandbox/Projection from Product of Family is Injection iff Other Factors are Singletons, $\pr'_i$ is injective.

From Projection from Product of Family is Surjective, $\pr'_i$ is surjective. From Projection from Product Topology is Continuous, $\pr'_i$ is continuous.

From Projection from Product Topology is Open, $\pr'_i$ is open.

Thus, by definition, we have that $\pr'_i$ is a homeomorphism.

Consider the projection:
 * $\pr_i: X \to X_i$

Then for all $y \in Y_i$:

By equality of mappings, $\pr_i {\restriction_{Y_i} } = \pr'_i$

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