A Cartesian product \(\prod_{\alpha\in A} X_\alpha\) is equipped with canonical projection maps \(\pi_\beta\colon \prod_{\alpha\in A} X_\alpha \to X_\beta\) defined by \(\pi_\beta(x)=x_\beta\).

Remarks

Finite Cartesian products Link to heading

Consider the product space topology on \(\prod_{i=1}^{n} X_i\) (i.e. \(A\) is finite). Then

Infinite Cartesian products Link to heading

The same is true as for finite product spaces. However, one needs to be careful with the chosen topology on a arbitrary Cartesian product (see [1], Sec. Infinite Products starting at page 63)

References Link to heading

  1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7