Separated by a function

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: roughly speaking, neither overlapping nor touching. 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.

There are various ways in which two subsets of a topological space X can be considered to be separated.

The separation axioms are various conditions that are sometimes imposed upon topological spaces which can be described in terms of the various types of separated sets. As an example, we will define the T₂ axiom, which is the condition imposed on separated spaces. Specifically, a topological space is separated if, given any two distinct points x and y, the singleton sets {x} and {y} are separated by neighbourhoods.

Separated spaces are also called Hausdorff spaces or T₂ spaces. Further discussion of separated spaces may be found in the article Hausdorff space. General discussion of the various separation axioms is in the article Separation axiom.

Given a topological space X, it is sometimes useful to consider whether it is possible for a subset A to be separated from its complement. This is certainly true if A is either the empty set or the entire space X, but there may be other possibilities. A topological space X is connected if these are the only two possibilities. Conversely, if a nonempty subset A is separated from its own complement, and if the only subset of A to share this property is the empty set, then A is an open-connected component of X. (In the degenerate case where X is itself the empty set {}, authorities differ on whether {} is connected and whether {} is an open-connected component of itself.)

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