A semi-metric space (M,d) satisfies all of the conditions of a metric space except it need NOT satisfy $d(f,g)=0 \iff f=g$.
Give 3 different examples of semi-metric spaces which are NOT metric spaces.
I understand the difference between a semi-metric space and metric space but I can not come up with 3 concrete examples.
Do the definitions of open, closed, dense and connected all still make sense in a semi-metric space without any alterations from the metric space setting?
I believe that the definition of open will still make sense in a semi-metric space. But the definition of closed, dense, and connected will not.
Am I right or wrong?