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 Sets: Definitions, 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":

... 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 ... not enough 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

... y are topologically distinguishable if there exists an open

**set**that one point belongs to but the other point does not ... other hand, if the singletons {x} and {y} are

**separated**, then the points x and y must be topologically distinguishable ...

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—H.L. (Henry Lewis)

“Protoplasm, simple or nucleated, is the formal basis of all life. It is the clay of the potter: which, bake it and paint it as he will, remains clay, *separated* by artifice, and not by nature from the commonest brick or sun-dried clod.”

—Thomas Henry Huxley (1825–1895)