Originally Posted by billym
Plato has said this already, but you seem to have missed it: Some sets are neither open nor closed. In fact, some sets (e.g. R) are both!
Here are two very straightforward definitions:
if and only if for each accumulation point (or limit point)
if and only if for each
there is a neighborhood
You may also benefit from the following reminders:
is a neighborhood
if and only if there is
is an accumulation point
(or limit point) of
if and only if
with every neighborhood of
containing infinitely many elements of
Please note that this last definition is not necessarily correct outside the context of real analysis. In other branches you must use a more rigorous definition, but for our purposes the above will suffice.
Okay, so now you are armed with these four definitions. Use them carefully.
Consider for example . It is trivial that . Furthermore, since accumulation points of are in by definition, we know that is a closed set.
Notice also that for any accumulation point and , we have . So is an open set, too!
Thus we have an example of a set which is both open and closed. You can also find examples of a set which is neither open nor closed (e.g. ).
Hopefully that helps.