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, separated, sets, set":

Separation Axiom - Preliminary Definitions
... separation axioms themselves, 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 distinct points ... for subsets of a topological space to be disjoint we may want them to be separated (in any of various ways) ...
Separated Sets - Relation To Topologically Distinguishable Points
... and y 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 ...

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