Natural Basis of Product Topology/Lemma 2
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Lemma for Natural Basis of Product Topology
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$:
- $\ds X := \prod_{i \mathop \in I} X_i$
Let $\BB$ be the set of cartesian products of the form $\ds\prod_{i \mathop \in I} U_i$ where:
Then:
- $\forall B_1, B_2 \in \BB : B_1 \cap B_2 \in \BB$
Proof
Let $B_1, B_2 \in \BB$.
Let $B_1 = \ds \prod_{i \mathop \in I} U_i$ where:
Let $J_1$ be the finite set of indices such that:
- $J_1 = \set {j \in I : U_i \neq X_i}$
Let $B_2 = \ds \prod_{i \mathop \in I} V_i$ where:
Let $J_2$ be the finite set of indices such that:
- $J_2 = \set{j \in I : V_i \neq X_i}$
Then:
\(\ds B_1 \cap B_2\) | \(=\) | \(\ds \paren{\prod_{i \mathop \in I} U_i } \cap \paren{\prod_{i \mathop \in I} V_i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \prod_{i \mathop \in I} \paren{ U_i \cap V_i}\) | General Case of Cartesian Product of Intersections |
Now:
\(\ds \forall i \in I: \, \) | \(\ds U_i, V_i \in \tau_i\) | \(\leadsto\) | \(\ds U_i \cap V_i \in \tau_i\) | Definition of topology $\tau_i$ |
and
\(\ds \forall i \in I \setminus \paren {J_1 \cup J_2}: \, \) | \(\ds U_i \cap V_i\) | \(=\) | \(\ds X_i \cap X_i\) | Definition of $J_1$ and $J_2$ | ||||||||||
\(\ds \) | \(=\) | \(\ds X_i\) | Set Intersection is Idempotent |
From Finite Union of Finite Sets is Finite, $J_1 \cup J_2$ is finite.
Then:
Thus $B_1 \cap B_2 \in \BB$
$\blacksquare$