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:
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“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)
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)