Product Topology - Properties

Properties

The product space X, together with the canonical projections, can be characterized by the following universal property: If Y is a topological space, and for every i in I, f_i : Y → X_i is a continuous map, then there exists precisely one continuous map f : Y → X such that for each i in I the following diagram commutes:

This shows that the product space is a product in the category of topological spaces. If follows from the above universal property that a map f : Y → X is continuous if and only if f_i = p_i o f is continuous for all i in I. In many cases it is easier to check that the component functions f_i are continuous. Checking whether a map g : X→ Z is continuous is usually more difficult; one tries to use the fact that the p_i are continuous in some way.

In addition to being continuous, the canonical projections p_i : X → X_i are open maps. This means that any open subset of the product space remains open when projected down to the X_i. The converse is not true: if W is a subspace of the product space whose projections down to all the X_i are open, then W need not be open in X. (Consider for instance W = R2 \ (0,1)2.) The canonical projections are not generally closed maps (consider for example the closed set whose projections onto both axes are R \ {0}).

The product topology is also called the topology of pointwise convergence because of the following fact: a sequence (or net) in X converges if and only if all its projections to the spaces X_i converge. In particular, if one considers the space X = RI of all real valued functions on I, convergence in the product topology is the same as pointwise convergence of functions.

Any product of closed subsets of X_i is a closed set in X.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy to show for finite products, while the general statement is equivalent to the axiom of choice.