the second map corresponds to the projection given by the action of the quotient group [Z.sub.2] = O(d)/SO(d) on the

quotient space [S.sup.c.sub.n]([R.sup.d])/SO(d) (SO(d) is normal in O(d)).

Then the group G = O(p + 1, q + 1) acts as conformal diffeomorphisms on [S.sup.p] x [S.sup.q], and also on its

quotient space X = ([S.sup.p] x [S.sup.q])/[Z.sub.2] by identifying the direct product of antipodal points.

In Section 2 we recall all necessary definitions, and in Section 3 we consider two axioms, denoted by M and G, each not derivable from S4 and the other one, and for each of them we give necessary and sufficient conditions under which it is valid in a

quotient space of a finite CW-complex, a particular point topological space, and an excluded point topological space.

intuitively showed a method of finding the inverse operation in the

quotient space of fuzzy numbers based on the Mares equivalence relation [16, 17], which have the desired group properties for the addition operation [18-20] midpoint function.

The

quotient space theory and random theory are the use equivalent class state "granularity", described the concept with "granularity" again.

A binary classification of multilevels granulation searching algorithm, namely, establishing an efficient multigranulation binary classification searching model based on hierarchical

quotient space structure, is proposed in this paper.

The strong neutrosophic vector space (V(I)/W(I),+,.) over a neutrosophic field K(I) is called a strong neutrosophic

quotient space.

Thus all the results in paper [2] can be proved by using the specific norm(1) in the

quotient space X/<e>.

Then [S.sub.2] = {a} [union] ([[union].sub.n[member of][omega]][T.sub.n]) is a

quotient space of ([T.sub.0] [union] {a}) [direct sum] T by identifying each [b.sub.n] [member of] T to [a.sub.n][member of][T.sub.0].

The Brownian map is described as a

quotient space of the continuous random tree called the CRT, for an equivalence relation that is defined in terms of Brownian labels assigned to the vertices of the CRT (see Sections 3 and 4 below for a detailed discussion).

Obvious reasoning establishes the

quotient space [II.sub.i][c.sub.i]/[~.sub.i] as a quasi-measurement structure.

The theory, fuzzy set theory, rough set theory, a superset of the theory of

quotient space and interval calculation, a branch of soft computing science (LI, 2005).