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 very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“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)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)