Topology of Plane Sets of Points 121
The set S is called the space of the topology, and the collection of sets
Q is called a topology on S (or, a topology for S). To introduce a topology
on Sis to indicate what subsets of S are to be considered as open in S. A
topological space consists of two objects: a nonempty set Sand a topology
Q on S. However, when no confusion arises it is customary to refer to the
topological space S.
The first mathematicians to recognize the importance of topological
concepts were Riemann (1826-1866) and Poincare (1854-1912). The first
to elaborate a theory of abstract topological spaces were Frechet [7] and
Hausdorff [9].
There are other approaches to the concept of topological space: based on
the notion of closure (Kuratowski) or on the notion of the neighborhood
(Hausdorff). Other possible but less used approaches are based on the
notion of interior of a set and on that of a closed set.
Properties 1, 2, and 3 of Definition 2.46 are the same properties of
open subsets of a metric space stated in Theorem 2.2 Hence if (S, d) is
a metric space and n is the collection of all open subsets of S, then
(S,r2) is a topological space. Thus any metric space is a topological space
with the topology induced or generated by the metric. This is called the
usual topology on a metric space. On the other hand, if a topological
space ( S, Q) is such that a metric d on S exists possessing the prop-
erty that a set is inn iff it is open in (S,d), then the topological space
( S, n) is said to be metrizable. There are many examples of nonmetriz-
able topological spaces (see Example 5 below). Some conditions have been
given for a topological space to be metrizable (Urysohn and Tychonoff
theorems).
Examples- Let S be a nonempty set, and let n = P(S), i.e., the collection of
all subsets of S. This is called the discrete topology on S. Any topological
space whose topology is the discrete topology is said to be a discrete space.
2. Let S be a nonempty set, and let Q = {0, S}, so that n consists
only of the empty set and the whole space S. This topology is at the other
end of that of example 1. However, they coincide if S is a singleton. The
topology {0,S} is called the indiscrete (or trivial) topology on S.- Let S be an infinite set, and let n consists of the empty set together
with all subsets of S whose complements are finite. (S, r2) is easily seen to
be a topological space. The class Q is called the cofinite topology on S. - Let S be a topological space, and let a be a nonempty subset of S.
The relative topology on a is defined to be the class of all intersections
with a of the open sets in S. The set a with its relative topology is itself
a topological space (called a subspace of S).