Product topology
The product topology is the canonical way of defining a topology on the Cartesian product of spaces.
Let
and
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 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
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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
- Real coördinate space as products of
with the standard topology, e.g. . - Torus topology
Properties
- Continuous maps from the product topology are continuous in each argument
- Canonical projections are open
Footnotes
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2020, Topology: A categorical approach, pp. 30–31 ↩