What is the difference between the two terms "clôture" and "fermeture" in mathematical contexts? For example, in topology, is it more common (or more proper) to say "la clôture de l'ensemble A" or "la fermeture de l'ensemble A"? What about other contexts?
To frame the question properly, my questions are: is there any difference between the two terms? which one is more frequently used? does the usage depend on the context? and is there any rule of thumb?
1) In a topological space the closure of a subset is almost always called adhérence.
I have never seen clôture in place of adhérence and I might remember having seen fermeture, but that would be in fairly old literature.
2) Algebraic closure is clôture algébrique, and similarly we have clôture séparable, radicielle, intégrale,....
3) The word fermeture is used for a relative situation:
For example we might say la fermeture algébrique de $\mathbb R$ dans $\mathbb C(X)$est $\mathbb C$, meaning that the set of elements of $\mathbb C(X)$ algebraic over $\mathbb R$ is $\mathbb C$.
Similarly we may speak of:
la fermeture séparable de $k$ dans $K$, la fermeture intégrale de $A$ dans $B$, etc.