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49 relations: Alexandroff extension, Base (topology), Boolean algebra (structure), Compact space, Compactification (mathematics), Comparison of topologies, Countably compact space, Counterexamples in Topology, Direct limit, Discrete space, Extremally disconnected space, Filter (mathematics), First uncountable ordinal, First-countable space, Glossary of topology, Hausdorff space, Interval (mathematics), Isolated point, J. Arthur Seebach Jr., Limit ordinal, Limit point, Linear continuum, Long line (topology), Lower limit topology, Lynn Steen, Mathematics, Neighbourhood system, Net (mathematics), Normal space, Ordinal number, Paracompact space, Partially ordered space, Rational number, Real line, Real number, Second-countable space, Separable space, Sequence, Sequentially compact space, Subbase, Subspace topology, Successor ordinal, Topological space, Topology, Total order, Totally disconnected space, Tychonoff plank, Union (set theory), Zero-dimensional space.
Alexandroff extension
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.
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Base (topology)
In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.
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Boolean algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Compactification (mathematics)
In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space.
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Comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.
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Countably compact space
In mathematics a topological space is countably compact if every countable open cover has a finite subcover.
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Counterexamples in Topology
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem, topologists (including Steen and Seebach) have defined a wide variety of topological properties.
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Direct limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way.
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Discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.
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Extremally disconnected space
In mathematics, a topological space is termed extremally disconnected if the closure of every open set in it is open.
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Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set.
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First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable.
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First-countable space
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".
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Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology.
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Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.
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Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.
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Isolated point
In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).
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J. Arthur Seebach Jr.
J.
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Limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.
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Limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.
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Linear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.
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Long line (topology)
In topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer".
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Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of interesting properties.
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Lynn Steen
Lynn Arthur Steen (January 1, 1941 – June 21, 2015) was an American mathematician who was a Professor of Mathematics at St. Olaf College, Northfield, Minnesota in the U.S. He wrote numerous books and articles on the teaching of mathematics.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Neighbourhood system
In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x is the collection of all neighbourhoods for the point x.
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Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
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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.
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Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
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Paracompact space
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite.
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Partially ordered space
In mathematics, a partially ordered space (or pospace) is a topological space X equipped with a closed partial order \leq, i.e. a partial order whose graph \ is a closed subset of X^2.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base.
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Separable space
In mathematics, a topological space is called separable if it contains a countable, 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.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Sequentially compact space
In mathematics, a topological space is sequentially compact if every infinite sequence has a convergent subsequence.
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Subbase
In topology, a subbase (or subbasis) for a topological space with topology is a subcollection of that generates, in the sense that is the smallest topology containing.
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Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).
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Successor ordinal
In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that is maximally disconnected, in the sense that it has no non-trivial connected subsets.
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Tychonoff plank
In topology, the Tychonoff plank is a topological space defined using ordinal spaces that is a counterexample to several plausible-sounding conjectures.
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space.
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Redirects here:
Induced order topology, Left order topology, Orderable topological space, Ordinal space, Right order topology.
References
[1] https://en.wikipedia.org/wiki/Order_topology
