Topology MOC

Product topology

The product topology is the canonical way of defining a topology on the Cartesian product of spaces. Let be an arbitrary collection of topological spaces with cartesian product

and as projections. The product topology on is the coarsest topology on for which all projections are continuous.1 #m/def/topology Thus it has Topological subbasis

Further characterisations

Explicit

The explicit characterisation is a little clunky due to the quirks of uncountable cartesian products. The product topology may be defined with the following topological basis #m/thm/topology

Proof of basis

It follows from the first characterisation that the following forms a Topological subbasis

When this is completed to a Topological basis via finite intersections, one obtains the explicit characterisation above.

Universal property for the product topoloogy

For every topological space and function , then is continuous iff is continuous for all . #m/thm/topology

Proof

We will first prove that the product topology satisfies the universal property. Let be topological spaces and let be the cartesian product endowed with the product topology . Let be a topological space, and be a function. If is continuous, then so are the compositions of continuous functions for all . Now suppose is continuous for all . We use the method of Proving continuity with a subbasis. Let . Then for some and . Since is continuous, . Thus the preïmage of every subbasic open set is open, whence is continuous. Therefore is continuous iff is continuous for all .

Now let be a topology on satisfying the same universal property. In particular, let and . Then since is continuous for all , so is , wherefore is coarser than . Now let and . Since is continuous, so too is for all . But is the coarsest topology on such that is continuous for all , so .

Spaces constructed as products

Properties

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2020, Topology: A categorical approach, pp. 30–31 ↩