This is nice subject to study. It also has important applications in many areas of applied mathematics, such as physics, chemistry, economics, meteorology and computer science. Probably others too.
Let me say this - you can think of this as a "tutorial" if you must, though I would rather you didn't. It's intended more as a "discussion platform". Also, it will be rather superficial - if more depth is wanted, I may be able to supply it. Finally - I am human (well almost), so I make mistakes. PLEASE correct me if you spot any. So here goes....
We start with a set
of points. We can, if we must, think of these points as being concrete objects like numbers, farmyard animals or the things in my garage, but it's best if we don't - better to keep them as abstract little buggers.
Now the
powerset on
, generally written as
, is the set comprising all the subsets of
.
Read that again: it's important to notice that, whereas
is a set of
points,
is a
set of sets of points. Ugh!
Also important is the fact that, by definition, every set has at least 2 subsets -
itself and the empty set
.
So, for example, if
, then
.
A
topology 
on
is defined as
, satisfying the following axioms.
the intersection of any finite number of elements of
is in
;
union of an arbitrary number of elements of
is in
;
;
The "indivisible pair"
is called a
topological space.
Notice that one can always associate more than one topology to a given set, so let's dispose of two rather uninteresting topologies. The first is the case that
which is called the "discrete topology".
Even less interesting is the so-called "trivial" or "indiscrete" topology
.
Now the elements (sets, recall) in
are called the
open sets in
. The closed sets are found as follows.
Suppose
. Then the
complement of
in
is simply all of
except
. This is often written as
, but I will use
.
So, whenever
is open in
id est 
, then
is closed in
.
Let's return to our simple example. Let
and suppose that
are the open sets in
.
Then
are the closed sets in
.
Notice then that while
is closed and not open, and
is open and not closed, the sets
are both open and closed and the set
is neither open nor closed. This generalizes.
Wow!
