Rostock (mathematics)

The object sprout on topological space expresses the local properties of the object. In a sense, it can be said that this is a new object, which adopts only local properties of the object of its origin (most often, such objects are mappings ). It is obvious that different functions can set the same germ. In this case, all local properties (continuity, smoothness, etc.) of such functions coincide and it is sufficient to consider the properties not of the functions themselves, but only their germs. The important point is to introduce the concept of locality, therefore, germs are considered for objects on a topological space.

Formal definition

Let point be given $x$ ${\ displaystyle x}$ topological space $X$ ${\ displaystyle X}$ and two displays $f,\;g:X\to Y$ ${\ displaystyle f, \; g: X \ to Y}$ in any set $Y$ ${\ displaystyle Y}$ . Then they say that $f$ ${\ displaystyle f}$ and $g$ ${\ displaystyle g}$ ask the same sprout in $x$ ${\ displaystyle x}$ if there is a neighborhood $U$ ${\ displaystyle U}$ points $x$ ${\ displaystyle x}$ such a restriction $f$ ${\ displaystyle f}$ and $g$ ${\ displaystyle g}$ on $U$ ${\ displaystyle U}$ match up. I.e,

f|_{U}=g|_{U}

{\ displaystyle f | _ {U} = g | _ {U}}

(which means $\forall x'\in U,\;f(x')=g(x')$ ${\ displaystyle \ forall x '\ in U, \; f (x') = g (x ')}$ ).

Similarly, they say about two subsets $S,\;T\subset X$ ${\ displaystyle S, \; T \ subset X}$ : they define the same germ in $x$ ${\ displaystyle x}$ if there is a neighborhood $U$ ${\ displaystyle U}$ such that:

S\cap U=T\cap U.

{\ displaystyle S \ cap U = T \ cap U.}

Obviously, the task of identical germs at a point $x$ ${\ displaystyle x}$ there is an equivalence relation (on mappings or sets, respectively), and these equivalence classes are called germs (germs of a map or germs of a set). The equivalence relation is usually denoted. $f\sim _{x}g$ ${\ displaystyle f \ sim _ {x} g}$ or $S\sim _{x}T$ ${\ displaystyle S \ sim _ {x} T}$ .

Sprout of this mapping $f$ ${\ displaystyle f}$ at the point $x$ ${\ displaystyle x}$ usually denote $[f]_{x}$ ${\ displaystyle [f] _ {x}}$ . Similarly, the germ given by the set $S$ ${\ displaystyle S}$ denote $[S]_{x}$ ${\ displaystyle [S] _ {x}}$ .

[f]_{x}=\{g:X\to Y\mid g\sim _{x}f\}.

{\ displaystyle [f] _ {x} = \ {g: X \ to Y \ mid g \ sim _ {x} f \}.}

Sprout displaying a point $x\in X$ ${\ displaystyle x \ in X}$ exactly $y\in Y$ ${\ displaystyle y \ in Y}$ write $(X,\;x)\to (Y,\;y)$ ${\ displaystyle (X, \; x) \ to (Y, \; y)}$ , in this way $f$ ${\ displaystyle f}$ is an entire equivalence class of mappings, and under $f$ ${\ displaystyle f}$ it is commonly understood any representative mapping. It can also be noted that two sets are equivalent (they specify the same germ of sets) if their characteristic functions are equivalent (with respect to the germs of the mappings):

S\sim _{x}T\Longleftrightarrow \mathbf {1} _{S}\sim _{x}\mathbf {1} _{T}.

{\ displaystyle S \ sim _ {x} T \ Longleftrightarrow \ mathbf {1} _ {S} \ sim _ {x} \ mathbf {1} _ {T}.}

Literature

Mishachev N.M., Eliashberg Ya. M. Introduction to the h-principle.

Rostock (mathematics)

Formal definition

Literature

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