is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.

Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism ( actually, even up to inner isomorphism ), this choice makes no difference as long as the space X is path-connected.

* The real line is path-connected, and is one of the simplest examples of a geodesic metric space

For example, a path-connected topological space is simply connected if each loop ( path from a point to itself ) in it is contractible ; that is, intuitively, if there is essentially only one way to get from any point to any other point.

Analogously, given a path-connected topological space, there is a Galois connection between subgroups of the fundamental group and path-connected covering spaces of.

The space has a universal cover if it is connected, locally path-connected and semi-locally simply connected.

A key result of the covering space theory says that for a " sufficiently good " space X ( namely, if X is path-connected, locally path-connected and semi-locally simply connected ) there is in fact a bijection between equivalence classes of path-connected covers of X and the conjugacy classes of subgroups of the fundamental group π < sub > 1 </ sub >( X, x ).

As a homotopy theory, the notion of covering spaces works well when the deck transformation group is discrete, or, equivalently, when the space is locally path-connected.

In mathematics, the Seifert-van Kampen theorem of algebraic topology, sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space, in terms of the fundamental groups of two open, path-connected subspaces and that cover.

Most of the main theorems about covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be path-connected, locally path-connected, and semi-locally simply connected.

In topology, a topological space is called simply connected ( or 1-connected ) if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question ( see below for an informal discussion ).

A topological space X is called simply connected if it is path-connected and any continuous map f: S < sup > 1 </ sup > → X ( where S < sup > 1 </ sup > denotes the unit circle in Euclidean 2-space ) can be contracted to a point in the following sense: there exists a continuous map F: D < sup > 2 </ sup > → X ( where D < sup > 2 </ sup > denotes the unit disk in Euclidean 2-space ) such that F restricted to S < sup > 1 </ sup > is f.

Suppose we have a path-connected space X, covered by path-connected open subspaces A and B whose intersection is also path-connected.

For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg-Maclane space, or a K ( G, 1 ).

If is a fibration with fiber F, with the base B path-connected, and the fibration is orientable over a field K, then the Euler characteristic with coefficients in the field K satisfies the product property:

Compare with the fact that H < sub > 1 </ sub >( X ) is the abelianization of the fundamental group π < sub > 1 </ sub >( X ) when X is path-connected.

An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenever p: → X and q: → X are two paths ( i. e.: continuous maps ) with the same start and endpoint ( p ( 0 )

For example, the cohomology ring of a path-connected H-space with finitely generated and free cohomology groups is a Hopf algebra.

As with all covering spaces, the fundamental group of G injects into the fundamental group of H. If G is path-connected then the quotient group is isomorphic to K. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space.

Let H be a topological group and let G be a covering space of H. If G and H are both path-connected and locally path-connected, then for any choice of element e * in the fiber over e ∈ H, there exists a unique topological group structure on G, with e * as the identity, for which the covering map p: G → H is a homomorphism.

As the above suggest, if a group has a universal covering group ( if it is path-connected, locally path-connected, and semilocally simply connected ), with discrete center, then the set of all topologically groups that are covered by the universal covering group form a lattice, corresponding to the lattice of subgroups of the center of the universal covering group: inclusion of subgroups corresponds to covering of quotient groups.

For example, with a mapping from a simplicial complex X to another, Y, defined initially on the 0-skeleton of X ( the vertices of X ), an extension to the 1-skeleton will be possible whenever Y is sufficiently path-connected.

The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces is always an amalgamated free product of the fundamental groups of the spaces.

The orthogonal matrices whose determinant is + 1 form a path-connected normal subgroup of O ( n ) of index 2, the special orthogonal group SO ( n ) of rotations.

The first homology group therefore vanishes if X is path-connected and π < sub > 1 </ sub >( X ) is a perfect group.

It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point.

* X is locally path-connected if and only if f ( X ) is

the set of homotopy classes of those closed curves γ based at x whose lifts γ < sub > C </ sub > in C, starting at c, are closed curves at c. If X and C are path-connected, the degree of the cover p ( that is, the cardinality of any fiber of p ) is equal to the index ( π < sub > 1 </ sub >( C, c )) of the subgroup p < sub >#</ sub > ( π < sub > 1 </ sub >( C, c )) in π < sub > 1 </ sub >( X, x ).

A topological space for which there exists a path connecting any two points is said to be path-connected.

In particular, many authors require both spaces to be path-connected and locally path-connected.

Let p: C → X be a covering map where both X and C are path-connected.

Any space may be broken up into a set of path-connected components.

If H is a path-connected, locally path-connected, and semilocally simply connected group then it has a universal cover.

0 this means the mapping of the path-connected components ; if we assume both X and Y are connected we can ignore this as containing no information.

* Locally connected and Locally path-connected topological spaces

Being locally path connected, a manifold is path-connected if and only if it is connected.

For path-connected spaces, therefore, we can write π < sub > 1 </ sub >( X ) instead of π < sub > 1 </ sub >( X, x < sub > 0 </ sub >) without ambiguity whenever we care about the isomorphism class only.

Suppose that B is path-connected.

In order to move segments of paths around, by homotopy to form loops returning to a base point in, we should assume, and are path-connected and that isn't empty.

Experience is not seen, as it is in classical rationalism, as presenting us initially with clear and distinct objects simply located in space and registering their character, movements, and changes on the tabula rasa of an uninvolved intellect.

It is notable that at this time he was writing with admiration of Cimabue's and Poussin's way of filling space.

Fifth, we have just completed a year's experience with our new space law.

And this, of course, is exactly what Madison Avenue has been accused of doing albeit in a primitive way, with its `` hidden persuaders '' and what the space merchants accomplish with much greater sophistication and precision.

The space between them is filled with pit-run gravel or earth.

Now place 12 pieces 1/2'' '' sq. on this edge as we did before and space them with the 5-3/4'' '' long `` tappets '', as they are called.

However, the possible absence of a center of symmetry not only moves the hydrogen atom off Af, but also allows the oxygen atoms to become nonequivalent, with Af at Af and Af at Af ( space group Af ), where Af represents the oxygens on one side of the Af layers and Af those on the other side.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.

The authors set about answering this fundamental question through a detailed investigation of the patient's ability, tactually, ( 1 ) to perceive figure and ( 2 ) to locate objects in space, with his eyes closed ( or turned away from the object concerned ).

One had to manage the given subjects, three diverse recent events, so as to make them part of a classical frieze, -- that is, a pattern of large figures filling the space, with not much else, against a blank background.

Part 1, deals with the classification of crystalline substances by space groups and is not a numerical data compilation.

Since the earth is rotating and the unleveled gyro-stabilized platform is fixed with respect to a reference in space, an observer on the earth will see the platform rotating ( with respect to the earth ).

The sheriff was occupied with maneuvering the car around in a very narrow space.

Science is fully competent to deal with any element of experience which arises from an object in space and time.

While we are filling outer space with scientific successes, for many the `` inner '' space of their soul is an aching void.

So, for happy years, Helva scooted around in her shell with her classmates, playing such games as Stall, Power-Seek, studying her lessons in trajectory, propulsion techniques, computation, logistics, mental hygiene, basic alien psychology, philology, space history, law, traffic, codes: all the et ceteras that eventually became compounded into a reasoning, logical, informed citizen.

Below these lines is a wide space with a horizontal crack dividing it.

Several unusual applications, such as a nuclear battery or fuel for space ships with nuclear propulsion, have been proposed for the isotope < sup > 242m </ sup > Am, but they are as yet hindered by the scarcity and high price of this nuclear isomer.

To practice architecture means to offer or render services in connection with the design and construction of a building, or group of buildings and the space within the site surrounding the buildings, that have as their principal purpose human occupancy or use.

While Renaissance artists sought nature to find their style, the Mannerists looked first for a style and found a manner. In Mannerist paintings, compositions can have no focal point, space can be ambiguous, figures can be characterized by an athletic bending and twisting with distortions, exaggerations, an elastic elongation of the limbs, bizarre posturing on one hand, graceful posturing on the other hand, and a rendering of the heads as uniformly small and oval.

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with