TITLE: Necessity of being Hausdorff in the definition of compactness?
QUESTION [9 upvotes]: According to R Engelking - General Topology: 

A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every open cover ${\{U_s}\}_{s\in S}$ of the space $X$ there exists a finite set ${\{s_1,s_2, \dots , s_k}\}\subset S$ such that $X = U_{s_1}\cup U_{s_2}\cup \dots U_{s_k}$.* 

And $^*$ means its footnote:

The reader should be warned that some authors do not include the assumption that $X$ is a Hausdorff space in the definition of compactness.

Actually, I have seen non-inclusion of Hausdorffness in definition of compactness in C Adams & R Franzosa's Introduction to Topology. Pure and Applied!
My questions are: If Engelking's book is right,

Why Hausdorffness of a space as a requirement for compactness is omitted in (especially undergraduate) texts even though Hausdorffness is a concept that is taught in those books?!
Would someone please give an example of a space that is compact according to the definition without Hausdorffness, and that is not a Hausdorff space?

Thank you.    
EDIT - One definition must be correct; because compactness is equivalent to many other statements e.g. closeness+boundedness (?) ; Does any equivalence of compactness still hold if it is not Hausdorff?    
EDIT 2 - I think Engelking's definition is wrong in the last sentence since it must be $X \subset U_{s_1}\cup U_{s_2}\cup \dots U_{s_k}$ (?)

REPLY [3 votes]: Others have already explained the differences in usage of various conventions. My first course in topology was taught by Engelking so perhaps I can say something about his motivations for this choice. He had very specific standards for notation and terminology. In particular, if I recall correctly, one guideline was that notions that are used more often deserve shorter names. If Engelking decided to exclude hausdorffness from the definition of a compact space, then the phrase "compact Hausdorff" would occur much more often in his book than "compact" on its own and that would be a waste of words. Hence he made a more economical choice of including this condition in the definition.