product topology

Cylinder set, the natural open set of a product topology.

By the definition of the product topology, for each there are open sets and such that .

There is a natural topology on this space, called the product topology.

For more properties and equivalent formulations, see the separate entry product topology.

This stems directly from the fact that, in the product topology, almost all the factors in a basic open set are the whole space.

We define a topology on X by means of a product topology.

The product topology used to define the Baire space can be described more concretely in terms of trees.

Such a sequence is naturally endowed with a topology, the product topology.

The product topology is the initial topology with respect to the family of projection maps.

The product topology is sometimes called the Tychonoff topology.

These sets of finite sequences are referred to as cylinder sets in the product topology.

Here, G x G is viewed as a topological space by using the product topology.

The product topology is the coarsest topology for which all the projections are continuous.

One can show that the topology induced by the metric is the same as the product topology in the above definition.

The definition of the product topology leads to this characterization of basic open sets:

The product in Top is given by the product topology on the Cartesian product.

Not every compact space is sequentially compact; an example is given by 2, with the product topology .

For example, in finite products, a basis for the product topology consists of all products of open sets.

The subspace topology and product topology constructions are both special cases of initial topologies.

We endow V with the discrete topology and X with the product topology.

Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables.

We may give R the product topology, where each copy of R is given the discrete topology.

While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties.

Locally at least this map looks like a projection map in the sense of the product topology, and is therefore open and surjective.

The projection map is continuous (with respect to the product topology on I x I) and is surjective.