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A [[Characteristic class|characteristic class]] with values in  $  H  ^  \star  (  ;  \mathbf Z _ {2} ) $, 
defined for real vector bundles. The Stiefel–Whitney classes are denoted by  $  w _ {i} $, 
$  i > 0 $, 
and for a real vector bundle  $  \xi $
over a topological space  $  B $, 
the class  $  w _ {i} ( \xi ) $
lies in  $  H  ^ {i} ( B;  \mathbf Z _ {2} ) $. 
These classes were introduced by E. Stiefel [[#References|[1]]] and H. Whitney [[#References|[2]]] and have the following properties. 1) For two real vector bundles  $  \xi , \eta $
over a common base,

$$ 
w _ {k} ( \xi \oplus \eta )  =  \sum _ { i } w _ {i} ( \xi ) w _ {k-} i ( \eta ),\ \ 
w _ {0}  =  1;
$$

in other words,  $  w( \xi \oplus \eta ) = w( \xi ) w( \eta ) $, 
where  $  w = 1+ w _ {1} + w _ {2} + \dots $
is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle  $  \zeta _ {1} $
over  $  \mathbf R P  ^  \infty  $
the equality  $  w( \zeta _ {1} ) = 1 + y $
holds, where  $  y $
is the non-zero element of the group  $  H  ^ {1} ( \mathbf R P  ^  \infty  ;  \mathbf Z _ {2} ) = \mathbf Z _ {2} $. 
These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e.  $  w( \xi \oplus \theta ) = w ( \xi ) $, 
where  $  \theta $
is the trivial bundle, and  $  w _ {i} ( \xi ) = 0 $
for  $  i >  \mathop{\rm dim}  \xi $. 
For an oriented  $  n $-
dimensional vector bundle  $  \xi $
over a base  $  B $, 
$  w _ {n} ( \xi ) \in H  ^ {n} ( B;  \mathbf Z _ {2} ) $
coincides with the reduction modulo 2 of the [[Euler class|Euler class]].

For a vector bundle  $  \xi $
over  $  B $, 
let  $  B  ^  \xi  $
be the [[Thom space|Thom space]] of this bundle. Further, let  $  \Phi :  H  ^  \star  ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {} ^ {\star+ n } ( B  ^  \xi  ;  \mathbf Z _ {2} ) $
be the [[Thom isomorphism|Thom isomorphism]]. Then the complete Stiefel–Whitney class  $  w( \xi ) $
coincides with

$$ 
\Phi  ^ {-} 1 Sq \Phi ( 1)  \in  H  ^  \star  ( B;  \mathbf Z _ {2} ),
$$

where  $  Sq = 1 + Sq  ^ {1} + Sq  ^ {2} + \dots $
is the complete [[Steenrod square|Steenrod square]]. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.

Any characteristic class with values in  $  H  ^  \star  (  ;  \mathbf Z _ {2} ) $, 
defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings  $  H  ^ {\star\star} (  \mathop{\rm BO} _ {n} ;  \mathbf Z _ {2} ) $
and  $  H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} ) $
are rings of formal power series in the Stiefel–Whitney classes:

$$ 
H  ^ {\star\star} (  \mathop{\rm BO} _ {n} ;  \mathbf Z _ {2} )  =  \mathbf Z _ {2} [[ w _ {1} \dots w _ {n} ]],
$$

$$ 
H  ^ {\star\star} (  \mathop{\rm BO} ;  \mathbf Z _ {2} )  =  \mathbf Z _ {2} [[ w _ {1} ,\dots ]].
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Stiefel,   "Richtungsfelden und Fernparallelismus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087790/s08779037.png" />-dimensionalen Mannigfaltigkeiten"  ''Comm. Math. Helv.'' , '''8''' :  4  (1935–1936)  pp. 305–353</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Whitney,   "Topological properties of differentiable manifolds"  ''Bull. Amer. Math. Soc.'' , '''43'''  (1937)  pp. 785–805</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W. Milnor,   J.D. Stasheff,   "Characteristic classes" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.E. Stong,   "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.E. Steenrod,   "The topology of fibre bundles" , Princeton Univ. Press  (1951)</TD></TR></table>

====Comments====
The notation  $  H  ^ {\star\star} ( X;  G) $
denotes the product of the Abelian groups  $  H  ^ {n} ( X;  G) $, 
while  $  H  ^  \star  ( X;  G) $
is the direct sum; the notation  $  H  ^  \star  ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {}  ^ {\star+} n ( B  ^  \xi  ;  \mathbf Z _ {2} ) $
means that there is a graded homomorphism of degree  $  n $: 
$  H  ^ {m} ( B;  \mathbf Z _ {2} ) \rightarrow \widetilde{H}  {}  ^ {m+} n ( B  ^  \xi  ;  \mathbf Z _ {2} ) $. 
For the classifying spaces  $   \mathop{\rm BO} _ {n} $
and  $   \mathop{\rm BO} $
see [[Classifying space|Classifying space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Husemoller,   "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>