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 $[a^{\prime },b^{\prime }]\times [c^{\prime },d^{\prime }]$ where $a^{\prime }<b^{\prime }$ and $c^{\prime }<d^{\prime }$ are real numbers.

Product of Countably Many Spaces: Consider the product space $\prod _{n=1}^{\mathrm{\infty }}[0,1]$ 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.