In a general topological setting what can be said about an open quasi-compact set? Is it true that a subset of such a set is compact? What if that set is open? I ask because this came up in class today with someones solution to a problem (my class consists mainly of student presentations). I should have asked at the time, but thought about it too much and didn't have an opportunity. So the statement seemed to be that any subset of an quasi-compact open set is quasi-compact- though for the problem we were discussing the subset was also open (though it seemed like this wasn't really used). It feels like it should be easy to prove or disprove but I haven't been able to come up with anything. Thanks. (Note: Hausdorff is not assumed)
Edit: I must apologize I have been away. In all instances where I merely wrote compact I meant quasi-compact. The point being that there is no assumption on the Hausdorff property.