Definition
Given a set X and an indexed family (Yi)i∈I of topological spaces with functions
the initial topology τ on is the coarsest topology on X such that each
is continuous.
Explicitly, the initial topology may be described as the topology generated by sets of the form, where is an open set in . The sets are often called cylinder sets. If I contains just one element, all the open sets of are cylinder sets.
Read more about this topic: Initial Topology
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)