hedipaldi said:Are they the eventually constant only?
The box topology on R^ω, also known as the product topology, is a topological space formed by taking the Cartesian product of countably many copies of the real number line R, each with the standard topology. This topology is commonly used in mathematics to study infinite-dimensional spaces.
The box topology is different from other topologies on R^ω, such as the uniform topology or the topology of pointwise convergence, because it considers each coordinate independently and does not take into account any relationships between them. This can lead to different notions of convergence and open sets compared to other topologies.
The basic open sets in the box topology on R^ω are the sets of the form U = U1 x U2 x ... x Un x R x R x ..., where Ui is an open subset of R for each i. These sets form a basis for the topology and any open set can be written as a union of these basic open sets.
No, the box topology on R^ω is not metrizable. This means that there is no metric that induces the same topology on R^ω. This is because the box topology allows for sets that are not open in the uniform topology, which is the topology induced by a metric.
The box topology on R^ω has many applications in mathematics, physics, and engineering. It is commonly used in the study of function spaces, topological vector spaces, and infinite-dimensional optimization problems. It also has applications in dynamical systems, control theory, and mathematical finance.