122 Chapter^2- Let S = {a,b,c} and let n = {0, {a}, {b}, {a,b}, {a,b,c}}. Clearly
( S, n) is a topological space. However,. S is not metrizable. To see this,
suppose that there is a metric d on S such that n is the collection of open
sets relative to d. If 5 < d(c,a) and 5 < d(c,b), then Ne(c) = {c} is open
in (S, d), yet not a member of n.
In a general topological space neighborhoods are defined as follows: N ( x)
is a neighborhood x E S if there exists an open set A E n such that x E A
and AC N(x). In particular, any open set is a neighborhood of each of its
points. However, a neighborhood of x, as defined above, need not be open.
More generally, a subset N of S is called a neighborhood of a set }( C S
iff there is an open set A E n such that K c A c N.
In terms of neighborhoods convergence of a sequence { xn} in S to a
point x ES is easily defined. It is said that Xn---+ x iff every N(x) contains
all but a finite number of terms of { xn}· But uniqueness of the limit
cannot be demonstrated in a general topological space. It is possible to
do so in ~n with the usual topology, because in that space two distinct
points have disjoint neighborhoods. If a similar condition is imposed on
a topological space, the important uniqueness property of the limit of a
sequence can be derived. This leads to several types of separation axioms
and corresponding classes of topological spaces. Following Alexandroff and
Hopf [1], the following nomenclature is in current usage.
Definitions 2.47 T 0 -space. A topological space is called a T 0 -space iff for
each pair of distinct points of the space there is a neighborhood of one of
these points that does not contain the other point.
Ti -space. A topological space is ~alled a Ti -space (also, a Frech et space)
iff each of any two distinct points of the space has a neighborhood that
does not contain the other point.
T2-space. A topological space is called a Trspace (also, a Hausdorff
space or a separated space) iff any two distinct points of the space have
disjoint neighborhoods.
Regular space. A topological space is called a regular space iff for any
x E S and any closed set F C S not containing x there are disjoint open
sets A and B (in the topology of S) such that x EA and F C B.
T 3 -space. A topological space is called a T 3 -space iff it is a regular
Ti ·space.
Normal space. A topological space is called a normal space iff for any
two disjoint closed sets Fi and F 2 in S there are disjoint open sets A and
B such that Fi C A and F2 C B.
T4-space. A topological space is called a T 4 -space iff it is a normal
Ti-space.