Table of Contents
78 relations: Annulus (mathematics), Banach space, Base (topology), Boundary (topology), Cantor set, Charles Weibel, Clopen set, Closed set, Closure (topology), Comb space, Complement (set theory), Connected space, Contractible space, Convex set, Covering space, Cycle graph, Discrete space, Discrete two-point space, Discrete valuation ring, Disjoint sets, Disjoint union, Disk (mathematics), Embedding, Empty set, Equivalence class, Equivalence relation, Euclidean space, Euclidean topology, Felix Hausdorff, Finite topological space, Frigyes Riesz, General linear group, Genus (mathematics), Graph (discrete mathematics), Hausdorff space, Hilbert space, Homotopical connectivity, Homotopy, Hyperconnected space, Idempotence, If and only if, Intermediate value theorem, Interval (mathematics), Local ring, Locally connected space, Long line (topology), Lower limit topology, Manifold, Mathematics, Maximal and minimal elements, ... Expand index (28 more) »
Annulus (mathematics)
In mathematics, an annulus (annuli or annuluses) is the region between two concentric circles.
See Connected space and Annulus (mathematics)
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
See Connected space and Banach space
Base (topology)
In mathematics, a base (or basis;: bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. Connected space and base (topology) are general topology.
See Connected space and Base (topology)
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of. Connected space and boundary (topology) are general topology.
See Connected space and Boundary (topology)
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties.
See Connected space and Cantor set
Charles Weibel
Charles Alexander Weibel (born October 28, 1950, in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
See Connected space and Charles Weibel
Clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. Connected space and clopen set are general topology.
See Connected space and Clopen set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. Connected space and closed set are general topology.
See Connected space and Closed set
Closure (topology)
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of. Connected space and closure (topology) are general topology.
See Connected space and Closure (topology)
Comb space
In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb.
See Connected space and Comb space
Complement (set theory)
In set theory, the complement of a set, often denoted by A^\complement, is the set of elements not in.
See Connected space and Complement (set theory)
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connected space and connected space are general topology and properties of topological spaces.
See Connected space and Connected space
Contractible space
In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Connected space and contractible space are properties of topological spaces.
See Connected space and Contractible space
Convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them.
See Connected space and Convex set
Covering space
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself.
See Connected space and Covering space
Cycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices (at least 3, if the graph is simple) connected in a closed chain.
See Connected space and Cycle graph
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, meaning they are isolated from each other in a certain sense. Connected space and discrete space are general topology.
See Connected space and Discrete space
Discrete two-point space
In topology, a branch of mathematics, a discrete two-point space is the simplest example of a totally disconnected discrete space.
See Connected space and Discrete two-point space
Discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
See Connected space and Discrete valuation ring
Disjoint sets
In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common.
See Connected space and Disjoint sets
Disjoint union
In mathematics, the disjoint union (or discriminated union) A \sqcup B of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come.
See Connected space and Disjoint union
Disk (mathematics)
In geometry, a disk (also spelled disc).
See Connected space and Disk (mathematics)
Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. Connected space and embedding are general topology.
See Connected space and Embedding
Empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
See Connected space and Empty set
Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.
See Connected space and Equivalence class
Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
See Connected space and Equivalence relation
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space.
See Connected space and Euclidean space
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
See Connected space and Euclidean topology
Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (à mon (Fr.).
See Connected space and Felix Hausdorff
Finite topological space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite.
See Connected space and Finite topological space
Frigyes Riesz
Frigyes Riesz (Riesz Frigyes,, sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators.
See Connected space and Frigyes Riesz
General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
See Connected space and General linear group
Genus (mathematics)
In mathematics, genus (genera) has a few different, but closely related, meanings.
See Connected space and Genus (mathematics)
Graph (discrete mathematics)
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related".
See Connected space and Graph (discrete mathematics)
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each that are disjoint from each other. Connected space and Hausdorff space are properties of topological spaces.
See Connected space and Hausdorff space
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional.
See Connected space and Hilbert space
Homotopical connectivity
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. Connected space and homotopical connectivity are general topology and properties of topological spaces.
See Connected space and Homotopical connectivity
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
See Connected space and Homotopy
Hyperconnected space
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). Connected space and hyperconnected space are properties of topological spaces.
See Connected space and Hyperconnected space
Idempotence
Idempotence is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application.
See Connected space and Idempotence
If and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.
See Connected space and If and only if
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval, then it takes on any given value between f(a) and f(b) at some point within the interval.
See Connected space and Intermediate value theorem
Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".
See Connected space and Interval (mathematics)
Local ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.
See Connected space and Local ring
Locally connected space
In topology and other branches of mathematics, a topological space X is locally connected if every point admits a neighbourhood basis consisting of open connected sets. Connected space and locally connected space are general topology and properties of topological spaces.
See Connected space and Locally connected space
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".
See Connected space and Long line (topology)
Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on \mathbb, the set of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of interesting properties.
See Connected space and Lower limit topology
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
See Connected space and Manifold
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Connected space and Mathematics
Maximal and minimal elements
In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S.
See Connected space and Maximal and minimal elements
Non-Hausdorff manifold
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. Connected space and Non-Hausdorff manifold are general topology.
See Connected space and Non-Hausdorff manifold
Number line
In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
See Connected space and Number line
Open set
In mathematics, an open set is a generalization of an open interval in the real line. Connected space and open set are general topology.
See Connected space and Open set
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.
See Connected space and Partition of a set
Path (topology)
In mathematics, a path in a topological space X is a continuous function from a closed interval into X. Paths play an important role in the fields of topology and mathematical analysis.
See Connected space and Path (topology)
Product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. Connected space and product topology are general topology.
See Connected space and Product topology
Projective module
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules.
See Connected space and Projective module
Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). Connected space and quotient space (topology) are general topology.
See Connected space and Quotient space (topology)
Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
See Connected space and Rational number
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Connected space and Real number
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.
See Connected space and Separated sets
Sierpiński space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed. Connected space and Sierpiński space are general topology.
See Connected space and Sierpiński space
Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Connected space and simply connected space are properties of topological spaces.
See Connected space and Simply connected space
Singleton (mathematics)
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element.
See Connected space and Singleton (mathematics)
Stack Exchange
Stack Exchange is a network of question-and-answer (Q&A) websites on topics in diverse fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process.
See Connected space and Stack Exchange
Subset
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
See Connected space and Subset
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). Connected space and subspace topology are general topology.
See Connected space and Subspace topology
Topological indistinguishability
In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. Connected space and topological indistinguishability are general topology.
See Connected space and Topological indistinguishability
Topological manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Connected space and topological manifold are properties of topological spaces.
See Connected space and Topological manifold
Topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Connected space and topological property are properties of topological spaces.
See Connected space and Topological property
Topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. Connected space and topological space are general topology.
See Connected space and Topological space
Topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
See Connected space and Topological vector space
Topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
See Connected space and Topologist's sine curve
Topology
Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
See Connected space and Topology
Totally disconnected space
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. Connected space and totally disconnected space are general topology and properties of topological spaces.
See Connected space and Totally disconnected space
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
See Connected space and Union (set theory)
Unit interval
In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.
See Connected space and Unit interval
Weak Hausdorff space
In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. Connected space and weak Hausdorff space are properties of topological spaces.
See Connected space and Weak Hausdorff space
References
Also known as 0-Connected, Arc Component, Arc connected, Arc connected space, Arc-connected, Arc-connected space, Arcwise connected, Arcwise-connected, Connected (topology), Connected component (topology), Connected set, Connected set in a topological space, Connected surface, Connected topological space, Connected topology, Connectedness (topology), Connex set, Disconnected (topology), Disconnected set, Disconnected space, Main theorem of connectedness, Path component, Path connected, Path connected space, Path connectedness, Path-connected, Path-connected component, Path-connected space, Path-connected topological space, Path-connectedness, Pathwise connected, Pathwise connected space, Pathwise-connected, Totally separated, Totally separated space.