Definition
Let X be a topological space and let C = {Cα : α ∈ A} be a family of subspaces of X (typically C will be a cover of X). Then X is said to be coherent with C (or determined by C) if X has the final topology coinduced by the inclusion maps
By definition, this is the finest topology on X for which the inclusion maps are continuous.
Equivalently, X is coherent with C if either of the following conditions holds:
- A subset U is open in X if and only if U ∩ Cα is open in Cα for each α ∈ A.
- A subset U is closed in X if and only if U ∩ Cα is closed in Cα for each α ∈ A.
Given a topological space X and any family of subspaces C there is unique topology on X which is coherent with C. This topology will, in general, be finer than the given topology on X.
Read more about this topic: Coherent Topology
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“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)
“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)