Skip to main content

Section Summary

Important ideas that we discussed in this section include the following. Throughout, let \((X_i, \tau_i)\) be topological spaces for \(i\) from \(1\) to some integer \(n\)
  • The product of the \(X_i\) is the Cartesian product \(\Pi_{i=1}^n X_i\text{.}\)
  • The set
    \begin{equation*} \CB = \left\{ \Pi_{i=1}^n O_i \mid O_i \text{ is open in } X_i\right\} \end{equation*}
    is a basis for the box topology on \(\Pi_{i=1}^n X_i\text{.}\)
  • The mapping \(\pi_j : \Pi_{i=1}^n X_i \to X_j\) defined by \(\pi_j((x_i)) = x_j\) is the projection map onto \(X_j\) for \(j\) from 1 to \(n\text{.}\)
  • A function \(f\) mapping a topological space \(Y\) to \(\Pi_{i=1}^n X_i\) is continuous if and only if \(\pi_j \circ f\) is continuous for every \(j\) from \(1\) to \(n\text{.}\)
  • Let \((X, \tau)\) be a topological space. A subset \(\CS\) of \(\tau\) is a subbasis for \(\tau\) if the set \(\CS\) of all finite intersections of elements of \(\CS\) is a basis for \(\tau\text{.}\)
  • If each \(X_i\) is (a) connected, (b) path connected, (c) compact, then \(\Pi_{i=1}^n X_i\) is (a) connected, (b) path connected, (c) compact with respect to the product topology.