# Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?

I’ve been trying to work out what generic means but I’m not making much progress.

You can find an example of the usage of the word generic for example here: “School on Generic Singularities in Geometry” or here:
“Rigidity of generic singularities of mean curvature flow”

Here is what I have so far:

“… “generic” singularities, i.e., singularities that appear for almost all maps …”

Just after that he mentions “generic maps” but does not elaborate on what it means. He mentions it later (kind of) by saying

“…The precise definition of a generic map is very delicate; for the
moment, we say only that any map may be approached by a generic map (up to an
approximation on the derivatives of order r), and that any map that is sufficiently close to
a generic map, in the preceding sense, is itself generic. …”

and later on he gives a definition of “generic singularity of a singular set”:

“… A critical point $x$ of $S_r$ will be called transversally critical, or, furthermore, generic, if the tangent planes to $\overline{f}(\mathbb R^n)$ and $F_r$ are in general position at the
point $\overline{f} (x)$ of the Grassmannian, which is assumed to be ordinary on $F_r$ . …”

Does this mean that generic and transversally critical are synonyms?

I feel I still don’t understand what generic means even after reading this definition and part of the article.

Please could someone help me understand what generic means?

#### Solutions Collecting From Web of "Mathematical definition of the word “generic” as in “generic” singularity or “generic” map?"

A property of points of topological space $X$ is generic if the property holds for all points on an open and dense set $Y\subset X$. Often this is mistaken for residual: A set is residual if it contains a countable intersection of open and dense sets. In many topological spaces residual sets are generic, by the Baire category theorem.

This is then often applied to function spaces. One constructs a space $C^\infty(X,\mathbb{R})$, with a topology, and shows that for a residual, or generic set $F\subset C^\infty(X,\mathbb{R})$ the property holds (for example the property of being a Morse function).