Box topology

In topology, the cartesian product of topological spaces can be given several different topologies. One of the more obvious choices is the box topology, where a base is given by the Cartesian products of open sets in the component spaces. Another possibility is the product topology, where a base is given by the Cartesian products of open sets in the component spaces, only finitely many of which can be not equal to the entire component space.
While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the component spaces are compact, the box topology on their Cartesian product will not necessarily be compact, although the product topology on their Cartesian product will always be compact. In general, the box topology is finer than the product topology, although the two agree in the case of finite direct products (or when all but finitely many of the factors are trivial).
Definition.
Given formula_1 such that
or the (possibly infinite) Cartesian product of the topological spaces formula_3, indexed by formula_4, the box topology on formula_1 is generated by the base
The name "box" comes from the case of R"n", in which the basis sets look like boxes.
Properties.
Box topology on R"ω":
Example - failure of continuity.
The following example is based on the Hilbert cube. Let R"ω" denote the countable cartesian product of R with itself, i.e. the set of all sequences in R. Equip R with the standard topology and R"ω" with the box topology. Define:
So all the component functions are the identity and hence continuous, however we will show "f" is not continuous. To see this, consider the open set 
Suppose "f" were continuous. Then, since:
there should exist formula_10 such that formula_11 But this would imply that 
which is false since formula_13 for formula_14 Thus "f" is not continuous even though all its component functions are.
Example - failure of compactness.
Consider the countable product formula_15 where for each "i", formula_16 with the discrete topology. The box topology on formula_1 will also be the discrete topology. Since discrete spaces are compact if and only if they are finite, we Immediately see that formula_1 is not compact, even though its component spaces are. 
formula_1 is not sequentially compact either: consider the sequence formula_20 given by 
Since no two points in the sequence are the same, the sequence has no limit point, and therefore formula_1 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 formula_1 with itself over an indexing set formula_24 is precisely the space of functions from formula_24 to formula_1"," denoted formula_27. The product topology yields the topology of pointwise convergence; sequences of functions converge if and only if they converge at every point of formula_24.
Because the box topology is finer than the product topology, convergence of a sequence in the box topology is a more stringent condition. Assuming formula_1 is Hausdorff, a sequence formula_30 of functions in formula_31 converges in the box topology to a function formula_32 if and only if it converges pointwise to formula_33 and 
there is a finite subset formula_34 and there is an formula_35 such that for all formula_36 the sequence formula_37 in formula_1 is constant for all formula_39. In other words, the sequence formula_37 is eventually constant for nearly all formula_41 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. 