Explicitly, the topology on *X* can be described as follows. A subset of *X* is open if and only if it is a union of (possibly infinitely many) intersections of finitely many sets of the form *p _{i}*

We can describe a basis for the product topology in a simple way using the bases of the constituting spaces *X _{i}*. Suppose that for each

If one starts with the standard topology on the real line **R** and defines a topology on the product of *n* copies of **R** in this fashion, one obtains the ordinary Euclidean topology on **R**^{n}.

The Cantor set is homeomorphic to the product of countably many copies of the discrete space {0,1} 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.

The product topology is also called the *topology of pointwise convergence* because of the following fact: a sequence (or net) in *X* converges if and only if all its projections to the spaces *X*_{i} converge. In particular, if one considers the space *X* = **R**^{I} of all real valued functions on *I*, convergence in the product topology is the same as pointwise convergence of functions.

An important theorem about the product topology is Tychonoff's theorem: any product of compact spaces is compact. This is easy for finite products, but the statement is (surprisingly) also true for infinite products, when the proof requires the axiom of choice in some form.

The product space *X*, together with the canonical projections, can be characterized by the following universal property: If *Y* is a topological space, and for every *i* in *I*, *f _{i}* :

To check whether a given map *f* : *Y* `->` *X* is continuous, one can use the following handy criterion: *f* is continuous if and only if *p _{i}* o

- Separation
- Every product of T
_{0}spacess is T_{0} - Every product of T
_{1}spacess is T_{1} - Every product of Hausdorff spaces is Hausdorff
- Every product of Regular spaces is Regular
- Every product of Tychonoff spaces is Tychonoff
- A product of normal spaces
*need not*be normal

- Every product of T
- Compactness
- Every product of compact spaces is compact (Tychonoff's theorem)
- A product of locally compact spaces
*need not*be locally compact

- Connectedness
- Every product of connected (resp. path-connected) spaces is connected (resp. path-connected)
- Every product of hereditarily disconnected spaces is hereditarily disconnected.