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Topological Property
In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... and related areas of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a topological property or topological invariant is a property of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... which is invariant under homeomorphism and a donut (torus In geometry, a torus (plural tori, ...
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Topology
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., topology (from the Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million as of ... words , and ) is concerned with the properties of a geometric object Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ... that are preserved under continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), t ...
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Regular Space
In topology and related fields of mathematics, a topological space ''X'' is called a regular space if every closed subset ''C'' of ''X'' and a point ''p'' not contained in ''C'' admit non-overlapping open neighborhoods. Thus ''p'' and ''C'' can be separated sets, separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms. Definitions A topological space ''X'' is a regular space if, given any closed set ''F'' and any Point (geometry), point ''x'' that does not belong to ''F'', there exists a neighbourhood (topology), neighbourhood ''U'' of ''x'' and a neighbourhood ''V'' of ''F'' that are Disjoint sets, disjoint. Concisely put, it must be possible to separated set, separate ''x'' and ''F'' with disjoint neighborhoods. A T3 space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a top ...
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Separable (topology)
In mathematics, a topological space is called separable if it contains a countable set, countable, dense (topology), dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff space, Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite set, finite or countably infinite is separable, for the w ...
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Isolated Point
] In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ... ''x'' is called an isolated point of a subset ''S'' (in a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood A neighbourhood (British English British Englis ...
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Discrete Space
In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ..., a discrete space is a particularly simple example of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has ...
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Perfectly Normal Hausdorff Space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. Definitions A topological space ''X'' is a normal space if, given any disjoint sets, disjoint closed sets ''E'' and ''F'', there are neighbourhood (topology), neighbourhoods ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be separated set, separated by neighbourhoods. A T4 space is a T1 space, T1 space ''X'' that is normal; this is equivalent to ''X'' being normal and Hausdorff space, Hausdorff. A completely normal space or a is a topological space ''X'' such that every s ...
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Precisely 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. Definitions There are various ways in which two subsets of a topological space ''X'' can be considered to be separated. * ''A'' and ''B'' are disjoint sets, disjoint if their intersection (set theory), intersection is the empty set. This property has nothing to do with topology as such, but only naive set theory, set theory. It is included here because it is the weakest in the sequence o ...
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Perfectly Normal Space
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. Definitions A topological space ''X'' is a normal space if, given any disjoint sets, disjoint closed sets ''E'' and ''F'', there are neighbourhood (topology), neighbourhoods ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be separated set, separated by neighbourhoods. A T4 space is a T1 space, T1 space ''X'' that is normal; this is equivalent to ''X'' being normal and Hausdorff space, Hausdorff. A completely normal space or a is a topological space ''X'' such that every s ...
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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: 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. Definitions There are various ways in which two subsets of a topological space ''X'' can be considered to be separated. * ''A'' and ''B'' are disjoint sets, disjoint if their intersection (set theory), intersection is the empty set. This property has nothing to do with topology as such, but only naive set theory, set theory. It is included here because it is the weakest in the sequence o ...
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Completely Normal
In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces. Definitions A topological space ''X'' is a normal space if, given any disjoint sets, disjoint closed sets ''E'' and ''F'', there are neighbourhood (topology), neighbourhoods ''U'' of ''E'' and ''V'' of ''F'' that are also disjoint. More intuitively, this condition says that ''E'' and ''F'' can be separated set, separated by neighbourhoods. A T4 space is a T1 space, T1 space ''X'' that is normal; this is equivalent to ''X'' being normal and Hausdorff space, Hausdorff. A completely normal space or a is a topological space ''X'' such that every s ...
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Partition Of Unity
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a partition of unity of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... ''X'' is a set ''R'' of continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...s from ''X'' to the unit interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
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Normal Space
In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... and related branches of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a normal space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... ''X'' that satisfies Axiom T4: every two disjoint closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "meas ...
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