Last semester we encountered the following definition of compactness in our first course on topology:
A topological space $X$ is said to be compact if every open cover of $X$ has a finite subcover.
This is also the definition found in Lee and Munkres. Now we have a first course on complex analysis, where we talked a bit about the topology on $\mathbb{C}$. The professor mentioned that he looked through the notes of our topology course and found a major mistake: The terminology of compactness is only established (or makes sense) for Hausdorff spaces, i.e. his definition would be (I also checked this in the literature, for example the definition occurs in general topology by Ryszard Engelking):
A topological space $X$ is said to be compact if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover.
This awoke my curiosity, so I looked up on wikipedia and found the following:
Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.
So my question is:
- Why is this difference regarding the terminology of compactness?
- Is it really that bad to not demand Hausdorffness of a compact space?