then the Cartesian product is the space of all real-valued functions on and convergence in the product topology is the same as pointwise convergence of functions. Examples If the real line is endowed with its standard topology then the product topology on the product of copies of is equal to the ordinary Euclidean topology on (Because is finite, this is also equivalent to the box topology on ) The Cantor set is homeomorphic to the product of countably many copies of the discrete space and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology. Several additional examples are given in the article on the initial topology. Properties The set of Cartesian products between the open sets of the topologies of each forms a basis for what is called the box topology on In general, the box topology is finer than the product topology, but for finite products they coincide. The product space together with the canonical projections, can be characterized by the following universal property: if is a topological space, and for every is a continuous map, then there exists continuous map such that for each the following diagram commutes. This shows that the product space is a product in the category of topological spaces. It follows from the above universal property that a map is continuous if and only if is continuous for all In many cases it is easier to check that the component functions are continuous. Checking whether a map is continuous is usually more difficult; one tries to use the fact that the are continuous in some way. In addition to being continuous, the canonical projections are open maps. This means that any open subset of the product space remains open when projected down to the The converse is not true: if is a subspace of the product space whose projections down to all the are open, then need not be open in (consider for instance ) The canonical projections are not generally closed maps (consider for example the closed set whose projections onto both axes are ). Suppose is a product of arbitrary subsets, where for every If all are then is a closed subset of the product space if and only if every is a closed subset of More generally, the closure of the product of arbitrary