It is well known that topology and algebraic geometry assign different meanings to the word "proper". Let us recall the relevant definitions from topology (and we work in the context of topological spaces):

A map $f:X\to Y$ is called

**separated**iff the diagonal $X\hookrightarrow X\times_YX$ is a closed inclusion.A map $f:X\to Y$ is called

**proper**iff for every net $(x_\alpha)_\alpha$ in $X$ such that $(f(x_\alpha))_\alpha$ converges to $y\in Y$, there exists a subnet converging to $x\in X$ with $f(x)=y$.~~Equivalently, for every compact subspace $K\subseteq Y$ its inverse image $f^{-1}(K)\subseteq X$ is compact (compact means every open cover admits a finite subcover)~~**EDIT:**The condition of compact sets having compact inverse images is*strictly weaker*than properness. Here is a counterexample: take the identity map $\{0,1\}\to\{0,1\}$ where the domain has the discrete topology and the target has the topology whose open sets are $\{\},\{0\},\{0,1\}$. The inverse image of a compact set under this map is compact (every subspace of $\{0,1\}$ with the discrete topology is compact), however this map is not proper in the sense of the definition above.

The above are the definitions from topology. In algebraic geometry, a map is called "proper" iff it is proper and separated in the sense defined above.

Is there a good reason to prefer one definition of the word "proper" over the other? Does the answer depend on whether one is doing topology or algebraic geometry?

**Note:** I am **not** asking for which meaning was first historically, nor for whether someone should be blamed for changing the meaning of an existing mathematical term, nor for people's personal preferences between the two meanings. Rather I am looking for a **mathematical** reason why one definition might be preferred over the other.

For example, both "separated" and "proper" are preserved under pullback, thus so is "separated and proper". But:

Is there another important formal property of "separated and proper" which

doesn't immediately followfrom corresponding formal properties of "separated" and "proper" individually?

Or, in the other direction:

Is there some important property of "proper" maps which cannot be formally derived from results about "separated" and "separated and proper" maps? Is there any argument using the notion of a "proper" map which

would become inconvenient if one only had access to the notions of "separated" and "separated and proper"?

For clarity, let me suggest that answers should specify whether they are using the word "proper" as I have defined it above or in the algebro-geometric sense (called "separated and proper" above).