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Countability conditions and convergent sequences
1. Second countable spaces
Definition 1. A space (X, T ) is called second countable if T has a countable base.
Examples 2. (1) R and R2 (with standard topology) are second countable.
(2) An uncountable discrete space is not second-countable.
Proposition 3. Any product of two second countable spaces is second countable.
Proposition 4. Any subspace of a second countable space is second countable.
Recall that the Sorgenfrey line is the real line with the topology S generated by
the base B = {[a, b) : a ∈ R, b ∈ Q, a < b}.
Example 5. The Sorgenfrey line is not second-countable.
Proof 1. The line y = −x in the subspace topology inherited from the Tychonoff
square of the Sorgenfrey line is discrete. Apply Example 2 (2), Propositions 3 and
4. 2
Proof 2. Follows from the next proposition. 2
Proposition 6. If a space (X, T ) has a countable base B then from other base B 0
of T one can extract a countable subfamily B 00 ⊂ B 0 such that B 00 is countable and
is still a base of T .
Proof. For all pairs B1 , B2 ∈ B for which it is possible fix a B(B1 , B2 ) such that
B1 ⊃ B(B1 , B2 ) ⊃ B2 . As B 00 , take the set of all such B(B1 , B2 ). 2
2. First countable spaces
Definition 7. Let (X, T ) be a topological space and x ∈ X. A family Bx ⊂ T is
called a local base of X at x if
(1) For every B ∈ Bx , x ∈ B;
(2) Whenever x ∈ U ∈ T , there is a B ∈ Bx such that x ∈ B ⊂ U .
The following theorem gives a way of constructing topologies from local bases.
Theorem 8. Let X be a set, and suppose for every x ∈ X a non empty family Bx
of subsets of X is fixed so that the following conditions are satisfied:
(1) For every B ∈ Bx , x ∈ B;
(2) For every B1 , B2 ∈ Bx , there is B ∈ Bx such that x ∈ B ⊂ B1 ∩ B2 ;
(3) Whenever x, y ∈ X, and y ∈ B ∈ Bx there is B 0 ∈ By such that y ∈ B 0 ⊂ B.
S
Then B = x∈X Bx is a base for a topology on X.
Definition 9. A space (X, T ) is first countable if for every x ∈ X there is a
countable local base Bx .
Examples 10. (1) The Sorgenfrey line is first countable (but not second countable).
(2) The Niemytzky plane is first countable but not second-countable.
1
2
(The Niemytzky plane, also called the Tangent Disc Upper Half-Plane, is the set
N = {hx, yi ∈ R2 : y ≥ 0} topologized as follows: for x ∈ R and y > 0 the local
base Bhx,yi consists of open discs around hx, yi of rational radius less than y; for a
point hx, 0i in the bottom line, the local base Bhx,yi consists of the sets of the form
{hx, 0i} ∪ Dr (x) where Dr (x) is an open disc of rational radius r tangent to the
bottom line from above at the point hx, 0i.)
Proposition 11. Every countable1 first countable space is second countable.
3. Convergent sequences
Let (X, T ) be a topological space and an ∈ X for all n ∈ N. The sequence
{an : n ∈ ω} may be treated as a mapping from N to X. Not necessarily all points
an are distinct.
Definition 12. The sequence {an : n ∈ ω} of points of X converges to a point
x ∈ X if every neighborhood2 of x in (X, T ) contains all but finitely many an .
Exercise 13. For the sequences given below, list all points to which the sequence
converges.
(1) an = n−1
n ∈ R where R bears the standard topology.
(2) The same sequence but R bears the Sorgenfrey topology.
(3) The same sequence but R bears the discrete topology.
(4) The same sequence but R bears the trivial3 topology.
(5) The same sequence but R bears the cofinite4 topology.
(6) an = n+1
n ∈ R where R bears the Sorgenfrey topology.
Definition 14. A topological space (X, T ) is called Hausdorff (or T2 ) if for every
distinct x, y ∈ X there exist U, V ∈ T such that x ∈ U , y ∈ V and U ∩ V = ∅.
Theorem 15. In a Hausdorff space, a sequence may converge at most to one point.
If x is the unique point to which the sequence {an : n ∈ N} converges, we write
lim an = x and say that x is the limit of {an : n ∈ N}.
n→∞
Recall that a space (X, T ) is T1 if for every x and for every y 6= x there is U ∈ T
such that x ∈ U but y 6∈ U . Equivalently, (X, T ) is T1 if all finite subsets of X are
closed.
Exercise 16. Give an example of a T1 space and a sequence in it which converges
to more than one point.
4. Fréchet spaces and sequential spaces
Definition 17. A space (X, T ) is Fréchet5 if whenever A ⊂ X and x ∈ A \ A,
there is a sequence of points of A that converges to x.
Theorem 18. If (X, T ) is Fréchet, A ⊂ X and x ∈ A, then there is a sequence of
points of A converging to x.6
1Saying that a space (X, T ) is countable we mean that X is countable.
2It is equivalent to say “every basic neighborhood”
3The one in which only the emptyset and the whole space are open
4The one in which only the whole space and all finite sets are closed
5Sometimes called Fréchet-Urysohn
6In the case when x ∈ A, it may be a trivial sequence, that is a sequence such that a = x for
n
all (or for all but finitely many) n.
3
In other words, if (X, T ) is Fréchet, and A ⊂ X, then A = {x ∈ X : there is a
sequence of points of A converging to x}.
Proposition 19. Every first countable space is Fréchet.
Proof. Let (X, T ) be first countable, and let A ⊂ X and x ∈ A \ A. Let
Bx = {Bn : n ∈ N} be a countable local base of X at x. For n ∈ N, pick
an ∈ A ∩ B1 ∩ ... ∩ Bn . Then the sequence {an : n ∈ ω} of points of A converges to
x. 2
Example 20. There is a (Hausdorff7) Fréchet space which is not first countable.8
Let X = {am,n : m, n ∈ N} ∪ {b} (where all points with different names are
distinct). X is topologized as follows. All points am,n are isolated. A basic neighborhood of the point b is determined by a function f : N → N where f may be
any such a function. So, let f : N → N and put Of (b) = {b} ∪ {am,n : m ∈ N and
n ≥ f (m)}. The set of all such Of (b) forms a local base of X at b. In other words,
a basic neighborhood of b consists of b itself, plus all but finitely many “first” points
of the set Am = {am,n : n ∈ N} for each m (you decide independently for each m
how many “first” points to remove from each Am ).9
Definition 21. A space (X, T ) is sequential if whenever A ⊂ X is not closed, there
is a point x ∈ A \ A and a sequence of points of A that converges to x.
Example 22. There is a (Hausdorff) sequential space which is not Fréchet.10
Let X = {am,n : m, n ∈ N} ∪ {bm : m ∈ N} ∪ {c} (where all points with
different names are distinct). X is topologized as follows. All points am,n are
isolated. A basic neighborhood of the point bm takes, for some N ∈ N the form
Bm,N = {bm } ∪ {am,n : n ≥ N }. A basic neighborhood of c is determined by a
number M ∈ N and a function f : N → N: put OM,f = {c}∪{bm : m ≥ M }∪{am,n :
m ≥ M and n ≥ f (m)}.11
Examples 23. There are (Hausdorff ) spaces which are not sequential.
(1) In the space X from Example 22, put A = {am,n : m, n ∈ N}, and consider
Y = A ∪ {c} with the subspace topology inherited from X. Then A \ A = {c}, but
there is no sequence converging from A to c.12
(2) Let D be an uncountable set, x 6∈ D, and L = D ∪ {x}. Topologize L as
follows: all points of D are isolated; a basic neighborhood of x takes the form
OC (x) = {x} ∪ (D \ C) where C is an arbitrary countable subset of D.13
5. Summary
2d countable ⇒ 1st countable ⇒ Fréchet ⇒ sequential
implications can be reversed
and neither of these
7Later we will see that this space, and some other spaces considered in this section are “much
better” than Hausdorff.
8The space described below is sometimes called the countable fan space
9We will discuss details in class
10The space described below is sometimes called the Arens space, or S
2
11We will discuss details in class
12We will discuss details in class
13This space is sometimes called the one-point Lindelöfication of D; we may discuss details in
class.