In reading Hartshorne,a topological space is quasi-compact if each open cover has a finite subcover(P80).Isn't it the definition for compactness of topological spaces?Am I right?Is quasi-compactness only in use in algebraic geometry in place of compactness?Or do we have another definition for compactness in algebraic geometry?Will someone be kind enough to say something on this?Thank you very much!
Hartshorne reserves "compact" for Hausdorff spaces, which many spaces in algebraic geometry fail to be. I'm not sure how prevalent this distinction is.
Before we use any terminology here, consider two conditions on a topological space $X$:
- Every open cover of $X$ has a finite subcover.
- For any two distinct points of $X$, there is an open set containing each such that the two open sets are disjoint. (This is the Hausdorff condition.)
Some texts use “$X$ is compact” to mean just (1); then “compact Hausdorff” is used to mean (1) and (2).
Other texts use “$X$ is compact” to mean both (1) and (2); then “quasi-compact” is used to mean just (1).
In the setting of algebraic geometry (e.g., the book by Hartshorne) just about every space under consideration satisfies (1). (This is because the topology is the Zariski topology, in which (a) it is pretty trivial that the whole space is closed and (b) one can show that every closed set has the property that every open cover has a finite subcover.) So in this context, not only will your space satisfy (1), but this fact is even pretty trivial. Therefore, a term that just indicates that (1) is satisfied will not be very useful in algebraic geometry.
For that reason, algebraic geometers will tend to follow the second convention, and use “compact” to mean that both conditions (1) and (2) are satisfied. But then there are situations in which they wish to indicate just (1), and so that's how you end up with instances of “quasi-compact” indeed appearing in books like Hartshorne to mean what others might think of as just “compact.”
By the way, the MO discussion quoted in the comment by user Ch Zh (namely https://mathoverflow.net/questions/16971/compact-and-quasi-compact) serves mostly as a discussion of this phenomenon for those who are already familiar with it, rather than as an explanation of the phenomenon for those who are unfamiliar with it. But confusion resulting from this distinction is somewhat inevitable; see, for example, this math.SE post as well: