In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets
Given a point $x$ of a topological space $X$, and two maps $f,\ g : X \rightarrow Y$ (where $Y$ is any set), then $f$ and $g$ define the same germ at $x$ if there is a neighbourhood $U$ of $x$ such that restricted to $U$, $f$ and $g$ are equal; meaning that $$f(u) = g(u)$$ for all $u$ in $U$. Similarly, if $S$ and $T$ are any two subsets of $X$, then they define the same germ at $x$ if there is again a neighbourhood $U$ of $x$ such that $S\cap U = T \cap U$.
It is straightforward to see that defining the same germ at $x$ is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written $f \sim_x g$ or $ S \sim_x T$.
Given a map $f$ on $X$, then its germ at $x$ is usually denoted $[f]_x$. Similarly, the germ at $x$ of a set $S$ is written $[S]_x$. Thus, $[f]_x = \{g:X\to Y \mid g \sim_x f\}$.
