Separated sets - Relation to separation axioms and separated spaces

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 aren't the same thing. And separable spaces are a completely differe ...

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• Separated sets - Definitions
• Separated sets - Relation to separation axioms and separated spaces
• Separated sets - Relation to connected spaces
• Separated sets - Relation to topologically distinguishable points

Read more here: » Separated sets: Encyclopedia - Separated sets

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 T ...

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Separated sets, Separated sets - Definitions, Separated sets - Relation to separation axioms and separated spaces, Separated sets - Relation to connected spaces, Separated sets - Relation to topologically distinguishable points

Read more here: » Separated sets: Encyclopedia II - Separated sets - Relation to separation axioms and separated spaces

There are various versions of the concept. The terms are defined below, where X is a topological space. First, two subsets A and B of X are disjoint if their intersection is the empty set. This property has nothing to do with topology as such, but only set theory; we include it here because it is the weakest in the sequence of different notions. For more ...

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Separated sets, Separated sets - Definitions, Separated sets - Relation to separation axioms and separated spaces, Separated sets - Relation to connected spaces, Separated sets - Relation to topologically distinguishable points

Read more here: » Separated sets: Encyclopedia II - Separated sets - Definitions

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 < ...

See also:

Separated sets, Separated sets - Definitions, Separated sets - Relation to separation axioms and separated spaces, Separated sets - Relation to connected spaces, Separated sets - Relation to topologically distinguishable points

Read more here: » Separated sets: Encyclopedia II - Separated sets - Relation to connected spaces