TheInfoList

In
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
, a
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 (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.

# Definition and equivalent formulations

A
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
''X'' is called ''simply connected'' if it is path-connected and any loop in ''X'' defined by ''f'' : S1 → ''X'' can be contracted to a point: there exists a continuous map ''F'' : D2 → ''X'' such that ''F'' restricted to S1 is ''f''. Here, S1 and D2 denotes the
unit circle measure. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, an ... and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: ''X'' is simply connected if and only if it is path-connected, and whenever ''p'' : ,1→ ''X'' and ''q'' : ,1→ ''X'' are two paths (i.e.: continuous maps) with the same start and endpoint (''p''(0) = ''q''(0) and ''p''(1) = ''q''(1)), then ''p'' can be continuously deformed into ''q'' while keeping both endpoints fixed. Explicitly, there exists a homotopy such that $F\left(x,0\right)=p\left(x\right)$ and $F\left(x,1\right)=q\left(x\right)$. A topological space ''X'' is simply connected if and only if ''X'' is path-connected and the fundamental group of ''X'' at each point is trivial, i.e. consists only of the
identity element In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. Similarly, is simply connected if and only if for all points $x,y\in X$, the set of
morphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s $\operatorname_\left(x,y\right)$ in the fundamental groupoid of has only one element. In
complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of com ...
: an open subset $X\subseteq\mathbb$ is simply connected if and only if both ''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'' 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 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. is simply connected because every loop can be contracted (on the surface) to a point. The definition rules out only Handle decomposition, 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 contractible space, contractibility.

# Examples

Image:Torus cycles.png, 150px, 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 R2 is simply connected, but R2 minus the origin (0,0) is not. If ''n'' > 2, then both R''n'' and R''n'' minus the origin are simply connected. * Analogously: the n-sphere, ''n''-dimensional sphere ''S''''n'' is simply connected if and only if ''n'' ≥ 2. * Every convex subset of Rn is simply connected. * A torus, the (elliptic) cylinder (geometry), cylinder, the Möbius 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, the special orthogonal group SO(''n'',R) is not simply connected and the special unitary group SU(''n'') is simply connected. * The one-point compactification of R is not simply connected (even though R is simply connected). * The long line (topology), long line ''L'' is simply connected, but its compactification, the extended long line ''L''* 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 (mathematics), genus (the number of ''handles'' of the surface) is 0. A universal cover of any (suitable) space ''X'' is a simply connected space which maps to ''X'' via a covering map. If ''X'' and ''Y'' are homotopy equivalent and ''X'' is simply connected, then so is ''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 - , which is not simply connected. The notion of simple connectedness is important in
complex analysis of the function . Hue represents the argument, brightness the magnitude. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of com ...
because of the following facts: * The Cauchy's integral theorem states that if ''U'' is a simply connected open subset of the complex number, complex plane C, and ''f'' : ''U'' → C is a holomorphic function, then ''f'' has an Antiderivative (complex analysis), antiderivative ''F'' on ''U'', and the value of every line integral in ''U'' with integrand ''f'' depends only on the end points ''u'' and ''v'' of the path, and can be computed as ''F''(''v'') - ''F''(''u''). The integral thus does not depend on the particular path connecting ''u'' and ''v''. * The Riemann mapping theorem states that any non-empty open simply connected subset of C (except for C itself) is conformal map, conformally equivalent to the unit disk. The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.