Basic Analysis – Gently Done Topological Vector Spaces

18 Basic Analysis
Theorem 2.13 Let X and Y be topological spaces. A mapping f : X → Y is
continuous if and only if whenever (x α ) I is a net in X convergent to x then the
net (f(x α )) I converges to f(x).
Proof Suppose that f : X → Y is continuous and suppose that (x α ) I is a net in
X such that x α → x. We wish to show that (f(x α )) I converges to f(x). To see
this, let V be any neighbourhood of f(x) in Y. Since f is continuous, there is a
neighbourhood U of x such that f(U) ⊆ V. But since (x α ) converges to x, it is
eventually in U, and so (f(x α )) I is eventually in V, that is, f(x α ) → f(x).
Conversely, suppose that f(x α ) → f(x) whenever x α → x. Let V be open in
Y. We must show that f −1 (V) is open in X. If this is not true, then there is
a point x ∈ f −1 (V) which is not an interior point. This means that every open
neighbourhood of x meets X \f −1 (V), the complement of f −1 (V). This is to say
that x is a limit point of X \f −1 (V) and so there is a net (x α ) I in X \ f −1 (V)
such that x α → x. By hypothesis, it follows that f(x α ) → f(x). In particular,
(f(x α )) I is eventually in V, that is, (x α ) I is eventually in f −1 (V). However, this
contradicts the fact that (x α ) I is a net in the complement of f −1 (V). We conclude
that f −1 (V) is, indeed, open and therefore f is continuous.
Nets can sometimes be useful in the study of the σ(X,F)-topology introduced
earlier.
Theorem 2.14 Let X be a non-empty set and let F be a collection of maps
f λ : X → Y λ from X into topological spaces (Y λ ,S λ ), for λ ∈ Λ. A net (x α ) I in X
converges to x ∈ X with respect to the σ(X,F)-topology if and only if (f λ (x α )) I
converges to f λ (x) in (Y λ ,S λ ) for every λ ∈ Λ.
Proof Suppose that x α → x with respect to the σ(X,F)-topology on X. Fix
λ ∈ Λ. Let V be any neighbourhood of fλ (x) in Yλ . Then, by definition of the
σ(X,F)-topology, f −1
λ (V) is a neighbourhood of x in X. Hence (xα ) I is eventually
in f −1
λ (V), which implies that (fλ (xα )) I is eventually in V. Thus fλ (xα ) → fλ (x)
along I, as required.
Conversely, suppose that (xα ) I is a net in X such that fλ (xα ) → fλ (x), for
each λ ∈ Λ. We want to show that (xα ) I converges to x with respect to the
σ(X,F)-topology. Let U be any neighbourhood of x with respect to the σ(X,F)topology.
Then there is m ∈ N, members λ1 ,...,λ m of Λ, and neighbourhoods
Vλ1 ,...,V λm of fλ1 (x),...,f (x), respectively, such that
λm
x ∈ f −1
λ1 (Vλ1 )∩···∩f−1 λm (V ) ⊆ U. λm
By hypothesis, for each 1 ≤ j ≤ m, there is β j ∈ I such that f λj (x α ) ∈ V λj
whenever α � β j . Let γ ∈ I be such that γ � β j for all 1 ≤ j ≤ m. Then
xα ∈ f −1
λ1 (Vλ1 )∩···∩f−1 λm (V ) ⊆ U λm
whenever α � γ. It follows that xα → x with respect to the σ(X,F)-topology.
Department of Mathematics King’s College, London