Anyway, one of the principal objects of study here is called a
neighbourhood. The definition couldn't be more simple:
Let
be a topological space. Then a neighbourhood of
is any open set containing
.
One writes
. Notice that usually, not always, each point will have more that one neighbourhood.
Now this might seem a little strange, but we are going to need a way to specify what exactly we mean when we say that 2 points in
are the same or different. To see why, notice that not every topological space has a metric. When it has, this is given by a map, say,
and one says that, whenever
In the case that no such simple metric is available we need an alternative definition. This is given us by the family of so-called "separation axioms", of which there are 5, called by the catchy names
. These are of increasing "stringency", in the sense that, if a space satisfies
it of necessity satisfies
etc.
Sane people only use
, which goes as follows.
If, for any 2 points
there exist neighbourhoods such that
, we will say that these 2 points are topologically distinguishable i.e. not equal in our topological space. And conversely.
Notice this crucial fact. We may have
and
, but this does NOT imply that
. In other words this property is not transitive, which reminds us of the triangle inequality for metric spaces.
But
guarantees there will always be neighbourhoods
.
A topological space with this property is called a
Hausdorff space.
So, to continue; roughly speaking, a space will be
connected if there is at least one continuous path between any 2 points. More precisely:
A topological space that
cannot be written as the union of 2 non-empty disjoint sets is said to be connected.
Alternatively, in a connected space, the only sets that are both open and closed are the base set and the empty set, say
and
. It is a fun exercise to bring these two definitions into register.
