James reduced product
Encyclopedia
In topology
, a branch of mathematics, the James reduced product J(X) of a topological space
X with given basepoint e is the quotient
of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words its underlying set is the free monoid generated by X (with unit e).
It was introduced by .
For a connected CW complex
X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X.
The commutative analogue of the James reduced product is called the infinite symmetric product
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics, the James reduced product J(X) of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X with given basepoint e is the quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words its underlying set is the free monoid generated by X (with unit e).
It was introduced by .
For a connected CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X.
The commutative analogue of the James reduced product is called the infinite symmetric product
Infinite symmetric product
In algebraic topology, the infinite symmetric product SP of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, .....
.