8. Product Spaces

Product Spaces

Product Topology and Its Properties

Product Topology: Let \( (X_i, \tau_i) \) be a collection of topological spaces indexed by \( i \) from some index set \( I \). The product topology on the Cartesian product \( \prod_{i \in I} X_i \) is defined to be the topology generated by the basis consisting of all sets of the form \( \prod_{i \in I} U_i \), where \( U_i \) is open in \( X_i \) for each \( i \in I \), and \( U_i = X_i \) for all but finitely many \( i \).

Properties of Product Topology:

• The product of two compact spaces is compact.
• The product of two connected spaces is connected.
• The product of two Hausdorff spaces is Hausdorff.

Tychonoff's Theorem

Tychonoff's Theorem: Tychonoff's theorem states that the product of any collection of compact spaces is compact.

Examples of Product Spaces

Product of Two Intervals: The product of two closed intervals \( [a, b] \times [c, d] \) in \( \mathbb{R}^2 \) with the standard topology is a product space. The product topology on this set is generated by the basis consisting of all rectangles \( [a', b'] \times [c', d'] \) where \( a' < b' \) and \( c' < d' \) are real numbers.

Product of Countably Many Spaces: Consider the product space \( \prod_{n=1}^\infty [0, 1] \) consisting of countably many copies of the unit interval. This space is compact by Tychonoff's theorem and can be identified with the Cantor set.