Separated Sets

In topology and related branches of mathematics, separated sets are pairs of subsets of a given topological space that are related to each other in a certain way. The notion of when two sets are separated or not is important both to the notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces.

Separated sets should not be confused with separated spaces (defined below), which are somewhat related but different. Separable spaces are again a completely different topological concept.

Read more about Separated SetsDefinitions, Relation To Separation Axioms and Separated Spaces, Relation To Connected Spaces, Relation To Topologically Distinguishable Points

Other articles related to "separated sets, set, sets, separated":

Separated Sets - Relation To Topologically Distinguishable Points
... are topologically distinguishable if there exists an open set that one point belongs to but the other point does not ... If x and y are topologically distinguishable, then the singleton sets {x} and {y} must be disjoint ... On the other hand, if the singletons {x} and {y} are separated, then the points x and y must be topologically distinguishable ...
Separation Axiom - Preliminary Definitions
... we give concrete meaning to the concept of separated sets (and points) in topological spaces ... But separated sets are not the same as separated spaces, defined in the next section.) The separation axioms are about the use of topological means to distinguish disjoint sets and ... subsets of a topological space to be disjoint we may want them to be separated (in any of various ways) ...

Famous quotes containing the words sets and/or separated:

    bars of that strange speech
    In which each sound sets out to seek each other,
    Murders its own father, marries its own mother,
    And ends as one grand transcendental vowel.
    Randall Jarrell (1914–1965)

    The new concept of the child as equal and the new integration of children into adult life has helped bring about a gradual but certain erosion of these boundaries that once separated the world of children from the word of adults, boundaries that allowed adults to treat children differently than they treated other adults because they understood that children are different.
    Marie Winn (20th century)