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 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

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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

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List of general topology topics - Compactness and countability. Compact space Relatively compact subspace Heine-Borel theorem Tychonoff's theorem Finite intersection property Compactification Measure of non-compactness Paracompact space Locally compact space Compactly generated space Axiom of countability First-countable space Second-countable space Separable space Lindel ...

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List of general topology topics, List of general topology topics - Basic concepts, List of general topology topics - Limits, List of general topology topics - Topological properties, List of general topology topics - Compactness and countability, List of general topology topics - Connectedness, List of general topology topics - Separation axioms, List of general topology topics - Topological constructions, List of general topology topics - Examples, List of general topology topics - Uniform spaces, List of general topology topics - Metric spaces, List of general topology topics - Topology and order theory, List of general topology topics - Descriptive set theory, List of general topology topics - Dimension theory, List of general topology topics - Topological algebra, List of general topology topics - Combinatorial topology, List of general topology topics - Foundations of algebraic topology

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Many of these names have alternative meanings in some of mathematical literature, as explained on History of the separation axioms; for example, the meanings of "normal" and "T4" are sometimes interchanged, similarly "regular" and "T3", etc. Many of the concepts also have several names. Most of these axioms have alternative definitions with the same meaning; the definitions given here are those which fall into a consistent pattern relating the various notions of separation defined in the previous section. Other ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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There are some other conditions on topological spaces that are sometimes classified with the separation axioms, but these don't fit in with the usual separation axioms as completely. Other than their definitions, they won't be discussed here. X is semiregular if the regular open sets form a base for the open sets of X. Any regular space must also be semiregular. X is fully normal if every open cover has an open star refinement. Every fully normal space must also be both normal regular and paracompact. In fact, fully normal spaces actually have more to do ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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The T0 axiom is special in that it cannot only be added to a property (so that regular plus T0 is T3) but also subtracted from a property (so that Hausdorff minus T0 is preregular), in a fairly precise sense; see Kolmogorov quotient for more information. When applied to the separation axioms, this leads to the relationships in the table below: In this table, you go from the right side to the left side by adding the requirement of T0, and you go from the left side to the right side b ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct; we may want them to be topologically distinguishable. Similarly, it's not enough for subsets of a topological space to be disjoint; we may want them to be separated (in any of various ways). The separation axioms all say, in one way or another, that points or sets that are di ...

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Separation axiom, Separation axiom - Separated sets and topologically distinguishable points, Separation axiom - Definitions of the axioms, Separation axiom - Relationships between the axioms, Separation axiom - Other separation axioms, Separation axiom - Sources

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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 < ...

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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 connected spaces

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