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.

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

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 ).

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.

For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.

For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

For example, upon rolling a 5-5 that player may move up to four separate checkers forward five spaces each.

For complete metric spaces this is equivalent to compactness.

For bassists with smaller hands, the large spaces between pitches may present a significant challenge, especially in the lowest range, where the spaces between notes are largest.

For example, if a child falls into a well and struggles to get out, he or she may develop a fear of wells, heights ( acrophobia ), enclosed spaces ( claustrophobia ), or water ( aquaphobia ).

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For example, they are Hausdorff paracompact spaces ( and hence normal and Tychonoff ) and first-countable.

For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology.

For a language like English, this is fairly trivial, since words are usually separated by spaces.

For locally compact spaces an integration theory is then recovered.

For natural lighting of interior spaces by admitting sunlight, see Daylighting.

For a given quadratic form, several different spaces of spinors with extra properties may exist.

For a brief period in fact, complete normed linear spaces where referred to as " Banach-Wiener " spaces in mathematical literature, based on terminology introduced by Wiener himself.

For instance, any isometry ( distance-preserving map ) between metric spaces is uniformly continuous.

For every vector space there exists a basis ( if one assumes the axiom of choice ), and all bases of a vector space have equal cardinality ( see dimension theorem for vector spaces ); as a result the dimension of a vector space is uniquely defined.

For example, algebraic topology makes use of Eilenberg – MacLane spaces which are spaces with prescribed homotopy groups.

For example, a topological group is just a group in the category of topological spaces.

For example, a corollary of the lemma is that normal T < sub > 1 </ sub > spaces are Tychonoff.

For the school year, 1959-1960, the Prince Edward County ( Virginia ) Board of Supervisors voted not to provide funds for public education, and the school board therefore could provide no public education -- for white or Negro children.

For Lamprias had said that the first articulate sound made is " alpha ", because it is very plain and simple — the air coming off the mouth does not require any motion of the tongue — and therefore this is the first sound that children make.

For the most part American vocabulary, phonology and syntax are used, to various extents, in Canada ; therefore many prefer to refer to North American English rather than American English.

For this reason, players should choose badminton shoes rather than general trainers or running shoes, because proper badminton shoes will have a very thin sole, lower a person's centre of gravity, and therefore result in fewer injuries.

For these and other reasons, BCG was therefore given to time with the peak incidence of pulmonary disease.

For example, he writes: " As, therefore, Jonah passed from the ship to the belly of the whale, so Christ passed from the cross to the sepulchre, or into the abyss of death.

For non-integer α, the functions J < sub > α </ sub >( x ) and J < sub >− α </ sub >( x ) are linearly independent, and are therefore the two solutions of the differential equation.

For approximately five days the armies therefore confronted each other across the plain of Marathon, in stalemate.

For clarity it should be stated that SDA understanding of Soul is not that of Calvin, by " soul " SDA theologians mean the physical person ( monism ), and that no component of human nature survives death, therefore each human will be " re_created " at resurrection.

For example, Georg Cantor ( who introduced this concept ) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers ( non-negative integers ), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

For many years, cardinal Giovanni di San Paolo ( elevated in 1193 ) was identified as member of the Colonna family and therefore its first representative in the College of Cardinals, but modern scholars have established that this was based on the false information from the beginning of 16th century.

For example, Spanish and French both come from Latin and therefore belong to the same family, the Romance languages.

For example, when a laboratory apparatus was developed that could reliably fire one electron at a time through the double slit, the emergence of an interference pattern suggested that each electron was interfering with itself, and therefore in some sense the electron had to be going through both slits at once — an idea that contradicts our everyday experience of discrete objects.

For Derrida the concept of neutrality is suspect and dogmatism is therefore involved in everything to a certain degree.

For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself.

For Orthodox no further council would therefore be needed until such time as a major crises arose within the church.

She therefore ( as mentioned at the Kypria according to Proclus as part of a plan hatched by Zeus and Themis ) tossed into the party the Apple of Discord, a golden apple inscribed Kallisti – " For the most beautiful one ", or " To the Fairest One " – provoking the goddesses to begin quarreling about the appropriate recipient.

For if k, m, and n are integers, and k is a common factor of two integers A and B, then A = nk and B = mk implies A − B = ( n − m ) k, therefore k is also a common factor of the difference.

For example, brosi ( to brush ) is based on a nominal root ( and therefore listed in modern dictionaries under the entry broso ), whereas kombi ( to comb ) is based on a verbal root ( and therefore listed under kombi ).

For example, gcd ( 42, 56 ) = 14, therefore,

For example, when MIT students surreptitiously put a fake police car atop the dome on MIT's Building 10, that was a hack in this sense, and the students involved were therefore hackers.

For example, until the Tannaitic period ( approximately 10 – 220 CE ) the months were set by observation of a new crescent moon, with an additional month added every two or three years to correct for the difference between twelve lunar months and the solar year, and therefore, to keep Passover in the spring.

For a fund structured as a company, shares will be both issued and redeemed at the net asset value (" NAV ") per share, so that if the value of the underlying investments has increased ( and the NAV per share has therefore also increased ) then the investor will receive a larger sum on redemption than it paid on investment.