Topologies on a fixed type α are ordered, by reverse inclusion. That is, for topologies t₁ and t₂ on α, we write t₁ ≤ t₂ if every set open in t₂ is also open in t₁. (One also calls t₁ finer than t₂, and t₂ coarser than t₁.)
Any function f : α → β induces induced f : topological_space β → topological_space α and coinduced f : topological_space α → topological_space β. Continuity, the ordering on topologies and (co)induced topologies are related as follows:
Topologies on α form a complete lattice, with ⊥ the discrete topology and ⊤ the indiscrete topology.
For a function f : α → β, (coinduced f, induced f) is a Galois connection between topologies on α and topologies on β.
There is a Galois insertion between topologies on α (with the inclusion ordering) and all collections of sets in α. The complete lattice structure on topologies on α is defined as the reverse of the one obtained via this Galois insertion.
finer, coarser, induced topology, coinduced topology
The open sets of the least topology containing a collection of basic sets.
The smallest topological space containing the collection g of basic sets
Construct a topology on α given the filter of neighborhoods of each point of α.
The inclusion ordering on topologies on α. We use it to get a complete lattice instance via the Galois insertion method, but the partial order that we will eventually impose on topological_space α is the reverse one.
If s equals the collection of open sets in the topology it generates, then s defines a topology.
The Galois insertion between set (set α) and topological_space α whose lower part sends a collection of subsets of α to the topology they generate, and whose upper part sends a topology to its collection of open subsets.
The complete lattice of topological spaces, but built on the inclusion ordering.
The ordering on topologies on the type α. t ≤ s if every set open in s is also open in t (t is finer than s).
Topologies on α form a complete lattice, with ⊥ the discrete topology and ⊤ the indiscrete topology. The infimum of a collection of topologies is the topology generated by all their open sets, while the supremem is the topology whose open sets are those sets open in every member of the collection.
A topological space is discrete if every set is open, that is, its topology equals the discrete topology ⊥.
Given f : α → β and a topology on β, the induced topology on α is the collection of sets that are preimages of some open set in β. This is the coarsest topology that makes f continuous.
Given f : α → β and a topology on α, the coinduced topology on β is defined such that s:set β is open if the preimage of s is open. This is the finest topology that makes f continuous.
This construction is left adjoint to the operation sending a topology on α to its neighborhood filter at a fixed point a : α.