Топологиялық кеңістіктердің ішкі жиындарының қатынастары

Type of relation for subsets of a topological space

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.

The separation axioms are various conditions that are sometimes imposed upon topological spaces, many of 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 usually called Hausdorff spaces or T2 spaces.

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.)

Given a topological space X, two points x 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. Thus for singletons, topological distinguishability is a condition in between disjointness and separatedness.