Then there exists a unique symplectic structure on \ mathcal { O } ( F ) such that inclusion map \ mathcal { O } ( F ) \ hookrightarrow \ mathfrak { g } ^ * is a moment map.
22.
The fact that a map is an embedding is often indicated by the use of a " hooked arrow ", thus : f : X \ hookrightarrow Y . On the other hand, this notation is sometimes reserved for inclusion maps.
23.
The homology " stabilizes " in the sense of stable homotopy theory : there is an inclusion map, and for fixed " k ", the induced map on homology is an isomorphism for sufficiently high " n ".
24.
Here " O " ( " X " ) is the partial order of open sets of " X " ordered by inclusion maps; and considered as a category in the standard way, with a unique morphism
25.
Inclusion maps are seen in algebraic topology where if " A " is a strong deformation retract of " X ", the inclusion map yields an isomorphism between all homotopy groups ( i . e . is a homotopy equivalence ).
26.
Inclusion maps are seen in algebraic topology where if " A " is a strong deformation retract of " X ", the inclusion map yields an isomorphism between all homotopy groups ( i . e . is a homotopy equivalence ).
27.
Two important special cases of the initial topology are the product topology, where the family of functions is the set of projections from the product to each factor, and the subspace topology, where the family consists of just one function, the inclusion map.
28.
In fact any attempt to define a model structure over some category of directed spaces has to face the following question : should an inclusion map \ { * \ } \ hookrightarrow [ 0, 1 ] be a cofibration, a weak equivalence, both ( trivial cofibration ) or none.
29.
The definition of ends given above applies only to spaces " X " that possess an exhaustion by compact sets ( that is, " X " must be direct system { " K " } of compact subsets of " X " and inclusion maps.
30.
Now the first unitary group U ( 1 ) is topologically a circle, which is well known to have a fundamental group isomorphic to "'Z "', and the inclusion map is an isomorphism on " ? " 1 . ( It has quotient the Stiefel manifold .)
