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Transfinite Induction and General Topology
A.3 Convergence
It is well known, see [Willard, 1970, Chapter 4], that sequences are not
adequate to describe all basic notions in topological spaces other than in the
class of first countable spaces (a topological space is called first countable if
it has a countable neighbourhood base at each of its points). One therefore
needs notions more general than that of sequence. Such generalizations are
provided by nets and filters, either of which is adequate to describe all topological concepts. Indeed, convergence itself can be taken as the fundamental
concept in developing topology, see Theorem 3.1.3, and this is the point of
view adopted in Chapter 3. However, we choose to work here only with nets
for reasons already mentioned in Chapter 3.
A.3.1 Definition A net in a set X is a mapping s : I → X, where (I, ≤) or
simply I is a directed set in which the ordering ≤ is reflexive and transitive.
For each i ∈ I, we denote s(i) by s i and denote the net s : I → X by (s i ) i∈I or
simply by (s i ) or just by s i if no confusion results. Similarly, sequences (s n ) n∈N ,
being special cases of nets, may be denoted simply by (s n ) or s n . Given a net
(s i ) i∈I in X and an element i 0 of I, we call the set (s i ) i0≤i = {s i | i 0 ≤ i}
a tail of (s i ) i∈I . A property will be said to hold eventually with respect to a
net (s i ) i∈I if it holds for some tail of the net.
A.3.2 Definition A subnet t of a net s : I → X is a net t : J → X satisfying
(i) t = s ◦ ϕ, where ϕ is a function mapping J into I, and (ii) for each i 0 ∈ I,
there exists j 0 ∈ J such that ϕ(j) ≥ i 0 whenever j ≥ j 0 . The point s ◦ ϕ (j)
is often denoted by s ij , and we refer to the subnet (s ij ) j∈J of (s i ) i∈I .
A.3.3 Definition Let X be a topological space, and let x ∈ X. A net (s i ) i∈I
in X will be said to converge to x, written s i → x or lim i s i = x, if, for each
neighbourhood U of x, there exists i 0 ∈ I such that s i ∈ U whenever i 0 ≤ i.
If s i → x, then we call x a limit of s i .
Since the singleton set {x} is a neighbourhood of x if X is endowed with
the discrete topology, it follows that s i → x in the discrete topology if and
only if (s i ) is eventually constant.
The notion of continuous function between topological spaces is fundamental in the subject. There are several ways of formulating this concept, but the
following is perhaps the most intuitive.
A.3.4 Definition Let X and Y be topological spaces, and suppose that f :
X → Y is a function. Then f is said to be continuous at x ∈ X if, for each
neighbourhood V of f (x) in Y , there is a neighbourhood U of x in X such
that f (U ) ⊆ V . We say f is continuous if it is continuous at x for each x ∈ X.
Transfinite Induction and General Topology
A.3 Convergence
It is well known, see [Willard, 1970, Chapter 4], that sequences are not
adequate to describe all basic notions in topological spaces other than in the
class of first countable spaces (a topological space is called first countable if
it has a countable neighbourhood base at each of its points). One therefore
needs notions more general than that of sequence. Such generalizations are
provided by nets and filters, either of which is adequate to describe all topological concepts. Indeed, convergence itself can be taken as the fundamental
concept in developing topology, see Theorem 3.1.3, and this is the point of
view adopted in Chapter 3. However, we choose to work here only with nets
for reasons already mentioned in Chapter 3.
A.3.1 Definition A net in a set X is a mapping s : I → X, where (I, ≤) or
simply I is a directed set in which the ordering ≤ is reflexive and transitive.
For each i ∈ I, we denote s(i) by s i and denote the net s : I → X by (s i ) i∈I or
simply by (s i ) or just by s i if no confusion results. Similarly, sequences (s n ) n∈N ,
being special cases of nets, may be denoted simply by (s n ) or s n . Given a net
(s i ) i∈I in X and an element i 0 of I, we call the set (s i ) i0≤i = {s i | i 0 ≤ i}
a tail of (s i ) i∈I . A property will be said to hold eventually with respect to a
net (s i ) i∈I if it holds for some tail of the net.
A.3.2 Definition A subnet t of a net s : I → X is a net t : J → X satisfying
(i) t = s ◦ ϕ, where ϕ is a function mapping J into I, and (ii) for each i 0 ∈ I,
there exists j 0 ∈ J such that ϕ(j) ≥ i 0 whenever j ≥ j 0 . The point s ◦ ϕ (j)
is often denoted by s ij , and we refer to the subnet (s ij ) j∈J of (s i ) i∈I .
A.3.3 Definition Let X be a topological space, and let x ∈ X. A net (s i ) i∈I
in X will be said to converge to x, written s i → x or lim i s i = x, if, for each
neighbourhood U of x, there exists i 0 ∈ I such that s i ∈ U whenever i 0 ≤ i.
If s i → x, then we call x a limit of s i .
Since the singleton set {x} is a neighbourhood of x if X is endowed with
the discrete topology, it follows that s i → x in the discrete topology if and
only if (s i ) is eventually constant.
The notion of continuous function between topological spaces is fundamental in the subject. There are several ways of formulating this concept, but the
following is perhaps the most intuitive.
A.3.4 Definition Let X and Y be topological spaces, and suppose that f :
X → Y is a function. Then f is said to be continuous at x ∈ X if, for each
neighbourhood V of f (x) in Y , there is a neighbourhood U of x in X such
that f (U ) ⊆ V . We say f is continuous if it is continuous at x for each x ∈ X.