Subspace of Product Space is Homeomorphic to Factor Space/Proof 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 $\ds \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$.
Suppose that $X$ is non-empty.
Then for each $i \in I$ there is a subspace $Y_i \subseteq X$ which is homeomorphic to $\struct {X_i, \tau_i}$.
Specifically, for any $z \in X$, let:
- $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$
and let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.
Then $\struct {Y_i, \upsilon_i}$ is homeomorphic to $\struct {X_i, \tau_i}$, where the homeomorphism is the restriction of the projection $\pr_i$ to $Y_i$.
Proof
Let $z \in X$.
Let $i \in i$.
Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$.
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
- $Y_i = \prod_{j \mathop \in I} Z_j$
$\Box$
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
- $\pr_i {\restriction_{Y_i} } = p_i$
$\Box$
Thus $\pr_i {\restriction_{Y_i} }: Y_i \to X_i$ is a homeomorphism.
$\blacksquare$