Download Convergence of Sequences and Nets in Metric and Topological

Convergence of Sequences and Nets in Metric and
Topological spaces
Thomas Wright
July 1, 2013
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Convergence and continuity in metric spaces
By convergence, on an abstract level, what we attempt to determine is, as we
take a walk through some kind of abstract space, is there some point to which
we are heading (even if we may never reach it), or are we simply aimlessly
wandering around. In order to clarify this notion, we must have first describe
the sort of paths we may walk along; in this vein we will start by defining a
sequence (e.g. of our steps):
Definition 1. A sequence in a set X is a function f : N → X. We write
xn = f (n) for all n ∈ N and denote the sequence as (xn )n∈N .
Now we may describe what it means to approach, or more formally converge
towards a point. If the set in which we wander is R, then our old friend Real
Analysis provides us with a definition,
Definition 2 (Convergence of real sequences). Let (xn )n∈N be a sequence with
terms in R. Now we say our sequence converges to a point a ∈ R if
∀ε ∈ R+ , ∃Nε ∈ N, ∀n ∈ N, n ≥ Nε ⇒ |xn − a| < ε
in which case we write,
(xn )n∈N → a
We can however extend this definition to an arbitrary metric space if our
one dimensional strolls on the real line start to seem a bit dull:
Definition 3 (Convergence of sequences in metric spaces). Let (X, d) be a
metric space and (xn )n∈N a sequence with terms in X. Then (xn )n∈N converges
to a point a ∈ X if
∀ε ∈ R+ , ∃Nε ∈ N, ∀n ∈ N, n ≥ Nε ⇒ xn ∈ Bε (a)
in which case we write,
(xn )n∈N → a
In metric spaces, the behaviour of sequences tells us a lot about the behaviour
of function. For example, considering continuity,
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Definition 4 (Continuity in Metric Spaces). Let f : X → Y be a function
between metric spaces (X, dX ) and (Y, dY ). f is continuous at a ∈ X if:
∀ε ∈ R+ , ∃δ ∈ R+ , f (Bδ (a)) ⊆ Bε (f (a))
we are able to determine exactly when a function is continuous on a metric
space (X, d) by considering how it treats convergent sequences in X,
Theorem 1 (Sequential characterisation of continuity). Let (X, dX ), (Y, dY ),
and f : X → Y a function between them. Then f is continuous at point a ∈ X
if and only if for all sequences (xn )n∈N with terms in X such that (xn )n∈N → a,
we have that (f (xn ))n∈N → f (a).
Proof. only if) Suppose f is continuous at a point a ∈ X and let (xn )n∈N be
a sequence such that (xn )n∈N → a. By convergence we may now pick δ ∈ R+
such that f (Bδ (a)) ⊆ Bε (f (a)) and Nδ ∈ N such that
n ≥ Nδ ⇒ xn ∈ Bδ (a) ⇒ f (xn ) ∈ f (Bδ (a)) ⊆ Bε (f (a))
proving (f (xn ))n∈N → f (a).
only if) Suppose f is discontinuous at a point a ∈ X. Then
∃ε ∈ R+, ∀δ ∈ R+, ∃x ∈ Bδ (a) , f (x) ∈
/ Bε (f (a))
Now for such an ε consider the sequence (xn )n∈N defined by picking xn ∈ B n1 (a)
such that f (xn ) ∈
/ Bε (f (a)) for all n ∈ N. By the Archimedian property of N
in R for ε ∈ R+ pick Nε ∈ N such that N1ε < ε, and so
n ≥ Nε ⇒ xn ∈ B n1 (a) ⊆ B N1 (a) ⊆ Bε (a)
ε
therefore (xn )n∈N → a. However, by the construction of (xn )n∈N , (f (xn ))n∈N 9
f (a).
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Convergence in topological spaces and the Hausdorff condition
For the more adventurous wanderer, who may choose to consider the convergence of sequences in topological spaces, we may further generalise the following
definition. For this task we will need to remember our notion of a neighbourhood:
Definition 5 (Neighbourhood). A neighbourhood G of a point x ∈ X in a
topological space (X, τ ) is any set G ∈ τ such that x ∈ G (that is, any open set
G in our topology containing x).
Definition 6 (Complete system of neighbourhoods). Let (X, τ ) be a topological
space. The the complete system of neighbourhoods N (x) of a point x ∈ X is
defined as
N (x) := {G ∈ τ : x ∈ τ }
(that is, the system of all neighbourhoods of our point x).
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Now we may proceed to give the general definition of convergence of a sequence in a topological space,
Definition 7 (Convergence of sequences in topological spaces). Let (X, τ ) be
a topological space and (xn )n∈N a sequence with terms in X. Then (xn )n∈N
converges to a point a ∈ X if
∀U ∈ N (a) , ∃NU ∈ N, ∀n ∈ N, n ≥ NU ⇒ xn ∈ U
in which case we write,
(xn )n∈N → a
Lest our experience with real sequences lead us to a sense of false complacency, the next example will demonstrate that sequence in topological spaces
do not behave quite as we would expect them to:
Example 1 (Convergence in the indiscrete topology). Consider the topological
space (X, {∅, X}), that is, the set X equipped with the indiscrete topology.
Then we see N (x) = {X} for all x ∈ X therefore for any sequence (xn )n∈N ⊆ X
and point X ∈ X we have that (xn )n∈N → x.
This shows us that limits of sequences in topological spaces need not converge
to a unique limit.
Our desire to know when sequences have unique limits of sequences motivates
us to introduce the Hausdorff condition:
Definition 8 (Hausdorff spaces). A topological space (X, τ ) satisfies the Hausdorff condition if for any distinct points a, b ∈ X there exist U ∈ N (a),
V ∈ N (b) such that U ∩ B = ∅ (that is, there exists a pair of disjoint neighbourhoods separating any pair of disjoint points).
We refer to a topological space satisfying the Hausdorff condition as a Hausdorff space.
Proposition 2 (Uniqueness of limits in Hausdorff spaces). In a Hausdorff space,
any given convergent sequence has a unique limit.
Proof. Suppose (X, τ ) is Hausdorff, (xn )n∈N is a sequence in X and (xn )n∈N →
a, (xn )n∈N → b. Now, supposing a 6= b there exist U ∈ N (a) , V ∈ N (b)
such that U ∩ V = ∅. Now by the convergence of our sequence, there exist
NU , NV ∈ D such that for all n ∈ N we have that n ≥ NU ⇒ xn ∈ U and
n ≥ NV ⇒ x n ∈ V .
Letting N = max{NU , NV } ∈ N we see xN ∈ U and xN ∈ V so xN ∈ U ∩ V
contrary to our assumption of disjointness proving a = b.
Also, unsurprisingly, the topologies of metric spaces are Hausdorff spaces,
Proposition 3. The topological space (X, τd ) induced by any metric space
(X, d) is Hausdorff.
Proof. For distinct points a, b ∈ X we have that r = d (x, y) 6= 0 and hence
B r2 (a) ∩ B r2 (b) = ∅ for B r2 (a) ∈ N (a) , B r2 (b) ∈ N (b) proving the Hausdorff
condition holds.
A further example will however show us that Hausdorffness does not coincide
exactly with the existence of unique limits to convergent sequences:
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Example 2 (Convergence in the cocountable topology). The Cocountable topology on a set X is defined as
τ := {G ∈ P : X \ G is countable} ∪ {∅}
Let (xn )n∈N be a sequence in X having limit y ∈ X. Now since N is
countable, so is {xn : n ∈ N}, so, (X \ {xn : n ∈ N}) ∪ {y} ∈ N (y). Therefore as (xn )y∈N converges to y we must have that n sufficiently large xn ∈
(X \ {xn : n ∈ N}) ∪ {y} and hence xn = y (that is, any convergent sequence
in the cocountable topology must eventually equal its limit). This proves the
uniqueness of limits.
We can however prove that if X is uncountable, (X, τ ) is not Hausdorff. Let
a, b ∈ X with a 6= b. Also let U ∈ N (a) and V ∈ N (b), then we know X \ U ,
X \ V and hence X \ (U ∩ V ) = (X \ U ) ∪ (X \ V ) are countable. Therefore U ∩ V
is uncountable and in particular U ∩ V 6= ∅ proving (X, τ ) is not Hausdorff.
Hence limits of sequences are unique in (X, τ ) despite it not necessarily being
Hausdorff.
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Nets and a fresh look at convergence
In order to better understand the properties of topological and Hausdorff spaces
we need to set aside our boring old sequences and introduced the more general
notion of a net. As we see when considering the cocountable topology, sequences
are rather limited by the fact that N is countable. Whilst this may not be an
issue which considering metric spaces (and more generally, sequential topological
spaces), this does not suffice for characterising properties of general topological
spaces. What N does however have going for it is the fact that it has an ordering
with certain nice properties, some of which we generalise in the following notion:
Definition 9 (Directed set). A directed set is a set D ordered by a preorder
relation ≤ (a reflexive, transitive binary relation), such that every two elements
in D have an upper bound in D (that is, ∀x, y ∈ D, ∃z ∈ D, x ≤ z ∧ y ≤ z).
Example 3 (Directed sets). We have the following useful directed sets:
1. The natural numbers N is a directed set as for all a, b ∈ N, z = max{a, b} ∈
N is such that a ≤ z and b ≤ z.
2. Any topology τ on a set X is a directed set with relation ⊆ as for all
U, V ∈ τ , W = U ∪ V ∈ τ is such that U ⊆ W and V ⊆ W .
3. The complete system of neighbourhoods N (a) of a point a is any topological space X is a directed set with relation ⊆.
Now we may proceed to define our generalised sequence, and consider what
it means for it to converge:
Definition 10 (Net). A net in a set X is a function f : D → X where D is a
directed set. We write xd = f (d) for all d ∈ D and denote the net as (xd )d∈D .
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Definition 11 (Convergence of nets in topological spaces). Let (X, τ ) be a
topological space and (xd )d∈D a net with terms in X. Then (xd )d∈D converges
to a point a ∈ X if
∀G ∈ N (a) , ∃dG ∈ D, ∀e ∈ D, e ≥ dG ⇒ xe ∈ G
in which case we write,
(xd )d∈D → a
Now we see that uniqueness of limits of nets exactly characterises Hausdorff
spaces:
Theorem 4. A topological space (X, τ ) is Hausdorff if and only if the limits of
all convergent nets with terms in X are unique.
Proof. only if) Suppose (X, τ ) is Hausdorff, (xd )d∈D is a net in X and (xd )d∈D →
a, (xd )d∈D → b. Now, supposing a 6= b there exist U ∈ N (a) , V ∈ N (b) such
that U ∩ V = ∅. Now by the convergence of our net, there exist dU , dV ∈ D
such that for all e ∈ D we have that e ≥ dU ⇒ xe ∈ U and e ≥ dV ⇒ xe ∈ V .
Since D is a directed set we may pick d ∈ D such that d ≥ dU and d ≥ dV ,
hence seeing xd ∈ U and xd ∈ V therefore xd ∈ U ∩V contrary to our assumption
of disjointness proving a = b.
if) We will proceed to prove the contrapositive. Supposing (X, τ ) is not
Hausdorff, then for some a, b ∈ X such that a 6= b, ∀U ∈ N (a) , ∀V ∈ N (b) , U ∩
V 6= ∅.
Now let
D := N (a) × N (b) = {(U, V ) : U ∈ N (a) , V ∈ N (b)}
which is a directed set under the preorder relation
(U1 , V1 ) ≤ (U2 , V2 ) ⇔ U1 ⊆ U2 ∧ V1 ⊆ V2
and consider the net (xd )d∈D defined by picking x(U,V ) ∈ U ∩ V for all d =
(U, V ) ∈ D.
Let U ∈ N (a), d = (U, X) ∈ D, then for all e = (W, Z) ∈ D, if e ≥ d then
W ≥ Z therefore xe = W ∩ Z ⊆ U ∩ X ⊆ U and hence (xd )d∈D → a.
Similarly (xd )d∈D → b proving (xd )d∈D has to distinct limits
The convergence of nets can also allows us to better understand the continuity of functions:
Definition 12 (Continuity of functions in topological spaces). Let (X, τ ), (Y, σ)
be topological spaces, and f : X → Y be a function between them. Then f is
continuous at a point a ∈ X if for every neighbourhood B ∈ σ of f (a), there
exists a neighbourhood A ∈ τ of a such that f (A) ⊆ B.
Theorem 5 (Characterisation of continuity by convergent nets). Let (X, τ ),
(Y, σ) be two topological spaces and f : X → Y be a function between them.
Then f is continuous at point a ∈ X if and only if for all nets (xd )d∈D with
terms in X such that (xd )d∈D → a, we have that (f (xd ))d∈D → f (a).
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Proof. only if) Suppose f is continuous and (xd )d∈D → a, then ∀V ∈ N (f (a)) , ∃U ∈
N (a) , f (U ) ⊆ V . Now by the convergence of (xd )d∈D we may pick dU ∈ D
such that
∀e ∈ D, e ≥ dU ⇒ xe ∈ U ⇒ f (xe ) ∈ f (U ) ⊆ V
proving that (f (xd ))d∈D → f (a) as required.
if) We will proceed to prove the contrapositive. Suppose f is discontinuous
at some a ∈ X. Then for some V ∈ N (f (a)), for all U ∈ N (a), f (U ) * V ,
that is, f (U ) \ V 6= ∅. Now consider the net (xU )U ∈N (a) , defined by picking
xU ∈ f −1 (f (U ) \ V ) ⊆ U for all U ∈ N (a). Now we can see (xd )d∈D → a
however the above shows (f (xd ))d∈D 9 a completing the proof.
References
[1] Wilson A. Sutherland. Introduction to Metric & Topological Spaces. 2009.
[2] Stijn Vermeeren. Sequences and nets in topology. July 2012.
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