This file introduces the following properties of a map f : X → Y between topological spaces:
(Open and closed maps need not be continuous.)
inducing f means the topology on X is the one induced via f from the topology on Y. These behave like embeddings except they need not be injective. Instead, points of X which are identified by f are also indistinguishable in the topology on X.
embedding f means f is inducing and also injective. Equivalently, f identifies X with a subspace of Y.
open_embedding f means f is an embedding with open image, so it identifies X with an open subspace of Y. Equivalently, f is an embedding and an open map.
closed_embedding f similarly means f is an embedding with closed image, so it identifies X with a closed subspace of Y. Equivalently, f is an embedding and a closed map.
quotient_map f is the dual condition to embedding f: f is surjective and the topology on Y is the one coinduced via f from the topology on X. Equivalently, f identifies Y with a quotient of X. Quotient maps are also sometimes known as identification maps.
open map, closed map, embedding, quotient map, identification map
A function between topological spaces is an embedding if it is injective, and for all s : set α, s is open iff it is the preimage of an open set.
A function between topological spaces is a quotient map if it is surjective, and for all s : set β, s is open iff its preimage is an open set.
An open embedding is an embedding with open image.
A closed embedding is an embedding with closed image.