In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.
This shape represents a set that is not simply connected, because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region.
A topological space is called simply connected if it is path-connected and any loop in defined by can be contracted to a point: there exists a continuous map such that restricted to is Here, and denotes the unit circle and closed unit disk in the Euclidean plane respectively.
An equivalent formulation is this: is simply connected if and only if it is path-connected, and whenever and are two paths (that is, continuous maps) with the same start and endpoint ( and ), then can be continuously deformed into while keeping both endpoints fixed. Explicitly, there exists a homotopy such that and
A topological space is simply connected if and only if is path-connected and the fundamental group of at each point is trivial, i.e. consists only of the identity element. Similarly, is simply connected if and only if for all points the set of morphisms in the fundamental groupoid of has only one element.[2]
In complex analysis: an open subset is simply connected if and only if both and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. It might also be worth pointing out that a relaxation of the requirement that be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components are simply connected.
Informal discussion
Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.
A sphere is simply connected because every loop can be contracted (on the surface) to a point.
The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.
Examples
A torus is not a simply connected surface. Neither of the two colored loops shown here can be contracted to a point without leaving the surface. A solid torus is also not simply connected because the purple loop cannot contract to a point without leaving the solid.
The Euclidean plane is simply connected, but minus the origin is not. If then both and minus the origin are simply connected.
The one-point compactification of is not simply connected (even though is simply connected).
The long line is simply connected, but its compactification, the extended long line is not (since it is not even path connected).
Properties
A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus (the number of handles of the surface) is 0.
A universal cover of any (suitable) space is a simply connected space which maps to via a covering map.
The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is which is not simply connected.
The notion of simple connectedness is important in complex analysis because of the following facts:
The Cauchy's integral theorem states that if is a simply connected open subset of the complex plane and is a holomorphic function, then has an antiderivative on and the value of every line integral in with integrand depends only on the end points and of the path, and can be computed as The integral thus does not depend on the particular path connecting and
Joshi, Kapli (August 1983). Introduction to General Topology. New Age Publishers. ISBN0-85226-444-5.
January 10, 2023
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In topology a topological space is called simply connected or 1 connected or 1 simply connected 1 if it is path connected and every path between two points can be continuously transformed intuitively for embedded spaces staying within the space into any other such path while preserving the two endpoints in question The fundamental group of a topological space is an indicator of the failure for the space to be simply connected a path connected topological space is simply connected if and only if its fundamental group is trivial Contents 1 Definition and equivalent formulations 2 Informal discussion 3 Examples 4 Properties 5 See also 6 ReferencesDefinition and equivalent formulations Edit This shape represents a set that is not simply connected because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region A topological space X displaystyle X is called simply connected if it is path connected and any loop in X displaystyle X defined by f S 1 X displaystyle f S 1 to X can be contracted to a point there exists a continuous map F D 2 X displaystyle F D 2 to X such that F displaystyle F restricted to S 1 displaystyle S 1 is f displaystyle f Here S 1 displaystyle S 1 and D 2 displaystyle D 2 denotes the unit circle and closed unit disk in the Euclidean plane respectively An equivalent formulation is this X displaystyle X is simply connected if and only if it is path connected and whenever p 0 1 X displaystyle p 0 1 to X and q 0 1 X displaystyle q 0 1 to X are two paths that is continuous maps with the same start and endpoint p 0 q 0 displaystyle p 0 q 0 and p 1 q 1 displaystyle p 1 q 1 then p displaystyle p can be continuously deformed into q displaystyle q while keeping both endpoints fixed Explicitly there exists a homotopy F 0 1 0 1 X displaystyle F 0 1 times 0 1 to X such that F x 0 p x displaystyle F x 0 p x and F x 1 q x displaystyle F x 1 q x A topological space X displaystyle X is simply connected if and only if X displaystyle X is path connected and the fundamental group of X displaystyle X at each point is trivial i e consists only of the identity element Similarly X displaystyle X is simply connected if and only if for all points x y X displaystyle x y in X the set of morphisms Hom P X x y displaystyle operatorname Hom Pi X x y in the fundamental groupoid of X displaystyle X has only one element 2 In complex analysis an open subset X C displaystyle X subseteq mathbb C is simply connected if and only if both X displaystyle X and its complement in the Riemann sphere are connected The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes a nice example of an unbounded connected open subset of the plane whose complement is not connected It is nevertheless simply connected It might also be worth pointing out that a relaxation of the requirement that X displaystyle X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement For example a not necessarily connected open set has a connected extended complement exactly when each of its connected components are simply connected Informal discussion EditInformally an object in our space is simply connected if it consists of one piece and does not have any holes that pass all the way through it For example neither a doughnut nor a coffee cup with a handle is simply connected but a hollow rubber ball is simply connected In two dimensions a circle is not simply connected but a disk and a line are Spaces that are connected but not simply connected are called non simply connected or multiply connected A sphere is simply connected because every loop can be contracted on the surface to a point The definition rules out only handle shaped holes A sphere or equivalently a rubber ball with a hollow center is simply connected because any loop on the surface of a sphere can contract to a point even though it has a hole in the hollow center The stronger condition that the object has no holes of any dimension is called contractibility Examples Edit A torus is not a simply connected surface Neither of the two colored loops shown here can be contracted to a point without leaving the surface A solid torus is also not simply connected because the purple loop cannot contract to a point without leaving the solid The Euclidean plane R 2 displaystyle mathbb R 2 is simply connected but R 2 displaystyle mathbb R 2 minus the origin 0 0 displaystyle 0 0 is not If n gt 2 displaystyle n gt 2 then both R n displaystyle mathbb R n and R n displaystyle mathbb R n minus the origin are simply connected Analogously the n dimensional sphere S n displaystyle S n is simply connected if and only if n 2 displaystyle n geq 2 Every convex subset of R n displaystyle mathbb R n is simply connected A torus the elliptic cylinder the Mobius strip the projective plane and the Klein bottle are not simply connected Every topological vector space is simply connected this includes Banach spaces and Hilbert spaces For n 2 displaystyle n geq 2 the special orthogonal group SO n R displaystyle operatorname SO n mathbb R is not simply connected and the special unitary group SU n displaystyle operatorname SU n is simply connected The one point compactification of R displaystyle mathbb R is not simply connected even though R displaystyle mathbb R is simply connected The long line L displaystyle L is simply connected but its compactification the extended long line L displaystyle L is not since it is not even path connected Properties EditA surface two dimensional topological manifold is simply connected if and only if it is connected and its genus the number of handles of the surface is 0 A universal cover of any suitable space X displaystyle X is a simply connected space which maps to X displaystyle X via a covering map If X displaystyle X and Y displaystyle Y are homotopy equivalent and X displaystyle X is simply connected then so is Y displaystyle Y The image of a simply connected set under a continuous function need not be simply connected Take for example the complex plane under the exponential map the image is C 0 displaystyle mathbb C setminus 0 which is not simply connected The notion of simple connectedness is important in complex analysis because of the following facts The Cauchy s integral theorem states that if U displaystyle U is a simply connected open subset of the complex plane C displaystyle mathbb C and f U C displaystyle f U to mathbb C is a holomorphic function then f displaystyle f has an antiderivative F displaystyle F on U displaystyle U and the value of every line integral in U displaystyle U with integrand f displaystyle f depends only on the end points u displaystyle u and v displaystyle v of the path and can be computed as F v F u displaystyle F v F u The integral thus does not depend on the particular path connecting u displaystyle u and v displaystyle v The Riemann mapping theorem states that any non empty open simply connected subset of C displaystyle mathbb C except for C displaystyle mathbb C itself is conformally equivalent to the unit disk The notion of simple connectedness is also a crucial condition in the Poincare conjecture See also EditFundamental group Mathematical group of the homotopy classes of loops in a topological space Deformation retract n connected space Path connected Unicoherent spaceReferences Edit n connected space in nLab ncatlab org Retrieved 2017 09 17 Ronald Brown June 2006 Topology and Groupoids Academic Search Complete North Charleston CreateSpace ISBN 1419627228 OCLC 712629429 Spanier Edwin December 1994 Algebraic Topology Springer ISBN 0 387 94426 5 Conway John 1986 Functions of One Complex Variable I Springer ISBN 0 387 90328 3 Bourbaki Nicolas 2005 Lie Groups and Lie Algebras Springer ISBN 3 540 43405 4 Gamelin Theodore January 2001 Complex Analysis Springer ISBN 0 387 95069 9 Joshi Kapli August 1983 Introduction to General Topology New Age Publishers ISBN 0 85226 444 5 Retrieved from https en wikipedia org w index php title Simply connected space amp oldid 1122243418, wikipedia, wiki, book, books, library,