a quasiorder being defined as reflexive and transitive - not antisymmetric (if also antisymmetric then it is apartial ordering).

it is aiff: (Ax,y)(x<=y). can someone provide me with a few canonical, if not pathological, examples of a trivial quasiorder?trivialquasiorder

what is the relation between aneighborhoodand anopen set? aspecialisation quasiorder.=. (x<=y) iff every neighborhood of x is a neighborhood of y. i need a few examples of a specialisation quasiorder.

why is a singleton necessarily a closed set -- why can't it be declared open?

thanks.