Continuity and constructions · Definition
Product topology
Before you start you need to know
It gives a natural topology to a pair of spaces at once, such as the plane built from two lines.
Definition
The topology on the product \(X\times Y\) of two spaces \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) that has as basis \(\mathcal{B}\) the sets \(U\times V\) with \(U\) and \(V\) open.
For two sets, the open sets of the product cannot be just the products \(U\times V\): the union of two of them is rarely another product. So these products are taken as a basis, and the open sets are all their unions. In \(\mathbb{R}\times\mathbb{R}\) this gives the open rectangles as a basis, and the usual topology of the plane is recovered.
The product topology is the coarsest one making both projections \(\pi_1:X\times Y\to X\) and \(\pi_2:X\times Y\to Y\) continuous: the preimage of an open set \(U\) is \(U\times Y\), which is basic. Moreover, a function into \(X\times Y\) is continuous exactly when both of its components are, which makes this the natural choice.
Two computational facts are worth remembering: the closure of a product is the product of the closures, \(\overline{A\times B}=\overline{A}\times\overline{B}\), and a product of bases is a basis of the product. The typical mistake is to believe that every open set of the product has the form \(U\times V\); in general it is only a union of such sets.
It builds on the notion of basis and prepares the study of compactness and connectedness of products, as well as the quotient topology, which glues points instead of multiplying spaces.
Think about it
Is the set \(\{(x,y):\ x>0\}\) open in \(\mathbb{R}^2\)?
Yes: it is \((0,\infty)\times\mathbb{R}\), a product of two open sets.
Why is the projection \(\pi_1\) continuous?
Because \(\pi_1^{-1}(U)=U\times Y\) is a product of open sets for every open \(U\) of \(X\).
Exercises 10
- Let \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\). How many distinct sets of the form \(U\times V\) are there, with \(U\) open in \(S\) and \(V\) open in \(T\)?
- With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), how many open sets does the product topology on \(S\times T\) have?
- With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), which of these subsets of \(S\times T\) is open?
- With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\) and the projection \(\pi_1:S\times T\to S\), how many points does \(\pi_1^{-1}(\{1\})\) have?
- Which statement about the open sets of the product topology on \(X\times Y\) is correct?
- With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), how many points does the closure of \(\{(0,a)\}\) in \(S\times T\) have?
- Which is a basis for the usual (product) topology on \(\mathbb{R}\times\mathbb{R}\)?
- In \(\mathbb{R}\times\mathbb{R}\) with the product topology, which of these sets is NOT open?
- With \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\) and \(T=\{a,b,c\}\) with open sets \(\varnothing,\{a\},\{a,b\},T\), how many open sets of \(S\times T\) are NOT of the form \(U\times V\)?
- Let \(S=\{0,1\}\) with open sets \(\varnothing,\{1\},S\). How many open sets does the product topology on \(S\times S\) have?
What comes next
References
- James R. Munkres, Topology, 2nd edition, Pearson, 2000. §15
- Stephen Willard, General Topology, Dover reprint, Dover Publications, 2004. §8