DR. SHEO KUMAR SINGH Assistant Professor, Department of Mathematics School of Physical Sciences, Mahatma Gandhi Central University, Motihari, Bihar-845401.

Email: [email protected] In my previous PPT, I have defined the product topology on the cartesian product 푋 × 푌, for topological spaces 푋 and 푌 in two ways which are given as follows:  the collection 픅 = 푈 × 푉 푈 is open in 푋 and 푉 is open in 푌} is a base for the product topology on 푋 × 푌. −1  the collection 풮 = {푝1 푈 | 푈 is open in 푋} ∪ −1 {푝2 푉 | 푉 is open in 푌} is a subbase for the product topology on 푋 × 푌 , where 푝1: 푋 × 푌 → 푋 and 푝2: 푋 × 푌 → 푌 defined respectively, by 푝1 푥, 푦 = 푥 and 푝2 푥, 푦 = 푦 are the projection maps in the components 푋 and 푌. −1 −1 Note: The basis element 푈 × 푉 = 푝1 푈 ∩ 푝2 푉 .

Here, I will generalize the above definition to a more general setting.

Consider the cartesian products 푋1 × ⋯ × 푋푛 and 푋1 × 푋2 × ⋯, where first is a finite product, and second is an arbitrary product of topological spaces 푋1, 푋2, … .

We can proceed in two ways:

 One way is to take all the sets of the form 푈1 × ⋯ × 푈푛 in the first case, and of the form 푈1 × 푈2 × ⋯ in the second case as basis elements, where 푈푖 is open in 푋푖.

 Another way is to generalize the subbase formulation, where we −1 take all the sets of the form 푝푖 푈푖 , where 푈푖 is open in 푋푖 as subbasis elements.

Here, we can observe that:

 In case of finite products, a typical basis element (i.e., basic open set) coincides in both the approaches and is equal to 푈1 × ⋯ × 푈푛.

 In case of an arbitrary product, a typical basis element is 푈1 × 푈2 × ⋯ using first approach while using the second approach, a typical basis −1 element is a finite intersection of subbasis elements 푝푖 푈푖 , say for 푖 = 푖1, … , 푖푟.

−1 Here note that 푝푖 푈푖 = 푋1 × ⋯ × 푋푖−1× 푈푖 × 푋푖+1 × ⋯ for all 푖. Thus, using the second approach, a typical basis element comes out to be 푈1 × 푈2 × ⋯, where 푈푖 = 푋푖 if 푖 ≠ 푖1, … , 푖푟, i,e., 푈푖 = 푋푖 except for finitely many values of 푖.

 Topology obtained using the first approach is said to be the box topology on the cartesian products 푋1 × ⋯ × 푋푛 and 푋1 × 푋2 × ⋯ .

 Topology obtained using the second approach is said to be the product topology on the cartesian products 푋1 × ⋯ × 푋푛 and 푋1 × 푋2 × ⋯ .

Thus, it is clear from the above discussion that the two topologies (i.e., product topology and box topology) are precisely the same for a finite product. But they differ for an arbitrary product and in this case the box topology is finer than the product topology.

Now the question is: Why we prefer the product topology in comparison to the box topology?

The answer to the above question is: There are several important theorems about finite products which will also hold for arbitrary products if we use the product topology, but not if we use the box topology.

For example, if 푋 and 푋푖′푠 are topological spaces, then a map 푓: 푋 → ∏푖∈퐼푋푖 is continuous if and only if 푝푖 ∘ 푓 is continuous for th all 푖 ∈ 퐼, where 푝푖 is the 푖 projection map.  James R. Munkres, Topology, 2nd ed., PHI.

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