Product projection
= Product projection
{title2=$\pi_i:\prod_{j\in I}X_j\to X_i$}
A product projection takes a tuple to its coordinate with the chosen index. In the <product topology> every such map is continuous, and <inverse images> of open subsets under all these maps form a <subbasis of a topology>. A map into the product is continuous exactly when its compositions with all product projections are continuous.