Since discrete spaces are compact if and only if they are finite, we immediately see that is not compact, even though its component spaces are. is not sequentially compact either: consider the sequence given by Since no two points in the sequence are the same, the sequence has no limit point, and therefore is not sequentially compact. Convergence in the box topology Topologies are often best understood by describing how sequences converge. In general, a Cartesian product of a space with itself over an indexing set is precisely the space of functions from to , denoted . The product topology yields the topology of pointwise convergence; sequences of functions converge if and only if they converge at every point of . Because the box topology is finer than the product topology, convergence of a sequence in the box topology is a more stringent condition. Assuming is Hausdorff, a sequence of functions in converges in the box topology to a function if and only if it converges pointwise to and there is a finite subset and there is an such that for all the sequence in is constant for all . In other words, the sequence is eventually constant for nearly all and in a uniform way. Comparison with product topology The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples—many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology. See also Cylinder set List of topologies Notes References Steen, Lynn A. and Seebach, J. Arthur Jr.; Counterexamples in Topology, Holt, Rinehart and Winston (1970). . External links Topological spaces Operations on structures