Question 1: The product topology is also called the topology of pointwise convergence because of the following fact: a ________ (or net) in X converges if and only if all its projections to the spaces X_{i} converge.  

Question 2: This is easy to show for finite products, while the general statement is equivalent to the ________.  

Question 3: A product of ________ need not be locally compact.  

Question 4: In topology and related areas of mathematics, a product space is the ________ of a family of topological spaces equipped with a natural topology called the product topology.  

Question 5: An important theorem about the product topology is Tychonoff's theorem: any product of ________ is compact.  

Question 6: If follows from the above universal property that a map f : Y → X is continuous ________ f_{i} = p_{i} o f is continuous for all i in I.  

Question 7: The ________ is equivalent to the statement that the product of a collection of nonempty sets is nonempty.  

Question 8: Every product of ________ is Hausdorff^{[1]}  

Question 9: A map that "locally looks like" a canonical projection F × U → U is called a ________.  

