Basic Analysis – Gently Done Topological Vector Spaces

2. Nets
In a metric space, a point belongs to the closure of a given set if and only if it is
the limit of some sequence of points belonging to that set. The convergence of the
sequence (a n ) n∈N to the point x is defined by the requirement that for any ε > 0
there is N ∈ N such that the distance between a n and x is less than ε whenever
n ≥ N. This is equivalent to the requirement that for any neighbourhood U of
x there is some N ∈ N such that a n belongs to U whenever n ≥ N. We take
this latter formulation, word for word, as the definition of convergence of the
sequence (a n ) n∈N in an arbitrary topological space. The resulting theory is not as
straightforward as one might suspect, as is illustrated by the following example.
Example 2.1 LetX betherealinterval[0,1]andletTbetheco-countabletopology
on X; that is, T consists of X and ∅ together with all those subsets G of X whose
complement X\G isacountable set. Let A = [0,1), and consider A. Now, {1} /∈ T
because X \{1} = [0,1) is not countable. It follows that the complement of {1}
is not closed. That is to say, A is not closed. However, the closure of A is closed
and contains A. This means that we must have A = [0,1], since [0,1] is the only
closed set containing A. Since 1 is not an element of A, it must be a limit point
of A.
Is there some sequence in A which converges to 1? Suppose that (a n ) n∈N is
any sequence in A whatsoever. Let B = {a 1 ,a 2 ,...} and let G = X \ B. Then
1 ∈ G. Since B is countable, it follows from the definition of the topology on X
that G is open. Thus G is an open neighbourhood of 1 which contains no member
of the sequence (a n ) n∈N . Clearly, (a n ) n∈N cannot converge to 1. No sequence in
A can converge to the limit point 1. We have exhibited a topological space with
a subset possessing a limit point which is not the limit of any sequence from the
subset.
This example indicates that sequences may not be as useful in topological
spaces as they are in metric spaces. One can, at this point, choose to abandon the
use of sequences, except perhaps in favourable situations, or to seek some form
of replacement. It turns out that one can keep the intuition of sequences but at
the cost of an increase in formalism. As perhaps suggested by the previous ex-
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