In the following picture two preference relations are defined -
Ist preference relation defines lexicographic relation. Though we know that lexicographic preferences doesn't have utility function. They are defined at a point so we cannot define a utility function - is discontinuous. But how can we show it in terms of upper and lower contour set being closed?
Why second one is continuous?
Note - some terminologies -
A binary relation >~ on the metric space X is continuous if, for all x in X, the upper and lower contour sets, { y in X : y > ~ x} and {y in X : x >~ y}, respectively, are closed.
The upper contour set of x is >~ (x) = {y in X : y >~ x}
The lower contour set of x is >~ (x) = {y in X : x>~ y}