Definition 3.1 A topology on a set X is a familyT of subsets of X, called open sets, satisfying the following requirements:
© Springer International Publishing Switzerland 2015
M. Manetti, Topology, UNITEXT - La Matematica per il 3+2 91, DOI 10.1007/978-3-319-16958-3_3
(A1) ∅ and X are open sets;
(A2) the union of any number of open sets is an open set; (A3) the intersection of two open sets is an open set.
A set equipped with a topology is called a topological space. Its elements are called
points.
ConditionA3 implies that any finite intersection of open sets is still an open set: in fact if A1, . . . , Anare open sets we can write A1∩· · ·∩ An= (A1∩· · ·∩ An−1)∩ An.
By induction on n the set A1∩ · · · ∩ An−1is open, so byA3 also A1∩ · · · ∩ Anis
open.
Any set admits at least one topology, typically several. For instance the family
T = P(X) of subsets of X is a topology called the discrete topology, while the family T containing only the empty set and X is a topology called trivial or indiscrete. Example 3.2 In the Euclidean topology onR, a subset U ⊂ R is open if and only
if it is the union of open intervals. Open sets thus defined satisfy A1, A2, A3 in Definition3.1: for example, if
A= i∈I ]ai, bi[ , B= h∈H ]ch, dh[ ,
are open sets, the distributive laws of union and intersection guarantee that
A∩ B = i∈I ]ai, bi[ ∩ h∈H ]ch, dh[ = i∈I,h∈H ]ai, bi[ ∩ ]ch, dh[
is still a union of open intervals. The Euclidean topology is often called the standard
topology.
Example 3.3 The upper topology onR is defined as the one having ] − ∞, a[, with a ∈ R ∪ {+∞} as non-empty open sets.
Definition 3.4 Let X be a topological space. A subset C ⊂ X is called closed if
X− C is open.
Since set-complementation swaps unions with intersections, the closed sets of a topology on X satisfy the following properties:
(C1) ∅ and X are closed;
(C2) any intersection of closed sets is closed; (C3) the union of two closed sets is closed.
As for open sets, conditionC3 implies that the finite union of closed sets is closed. It is clearly possible to describe a topology by telling what are its closed sets. For example, the cofinite topology (or finite-complement topology) on X decrees a subset C ⊂ X closed if and only if either C = X or C is finite.
3.1 Topological Spaces 41 Definition 3.5 LetT be a topology on a set X. A subfamily B ⊂ T is called a basis
ofT if every open set A ∈ T can be written as union of elements of B.
Example 3.6 Open intervals form a basis for the Euclidean topology on the real line.
A basis determines uniquely the topology, because any union of basis elements is open, and conversely any open set has that form.
Theorem 3.7 Let X be a set and B ⊂ P(X) a family of subsets. There exists a
topology on X for whichB is a basis if and only if two conditions hold: 1. X = ∪{B | B ∈ B};
2. for any pair A, B ∈ B and any point x ∈ A ∩ B there exists C ∈ B such that x∈ C ⊂ A ∩ B.
Proof The two conditions are clearly necessary, so let’s prove they are also sufficient.
Define open sets to be arbitrary unions of elements ofB. Then X is open (union of everything),∅ is open (empty union), and the union of open sets is easily open. The second condition implies that for every A, B ∈ B we have
A∩ B = ∪{C | C ∈ B, C ⊂ A ∩ B}.
If U = ∪Ai and V = ∪Bjare unions of elements Ai, Bj ∈ B, by the distributive
laws we obtain
U∩ V = ∪
i, j(Ai∩ Bj) = ∪{C | C ∈ B and ∃ i, j such that C ⊂ Ai ∩ Bj}.
A common mistake the reader should prevent is to mix the notion of a topological basis with Theorem3.7: in the definition the topology is given, and the criterion to be checked is not the one stated in the theorem, but that any open set can be written as union of open sets from the basis.
There is a natural order relation among the topologies of a given set.
Definition 3.8 LetT and R be topologies on the same set. One says that T is finer
(or stronger, larger) than R (and that R is coarser, weaker, or smaller than T ), writtenR ⊂ T , when every open set of R is open in T .
Example 3.9 The lower-limit line is defined as the set of real numbers with the
topology whose basis consists of the family of half-open intervals[a, b[. As ]a, b[ = ∪c>a[c, b[, this topology is finer that the Euclidean topology.1
Given an arbitrary collection {Ti} of topologies on a set X their intersection T = ∩iTi is a topology. If theTi are the only topologies containing a given family
S ⊂ P(X) of subsets of X, then T is the coarsest topology among those containing
elements ofS as open sets.
1Mathematicians do not invent strange-looking topologies just for fun: Robert Sorgenfrey, for
instance, defined the lower-limit topology to answer the question of whether the product of para- compact spaces (Definition7.14) is paracompact (it’s not [So47]).
Example 3.10 Let{Xi | i ∈ I } be a collection of topological spaces. Consider on
their disjoint union X =i Xi the coarsest topology among those containing the
topology of each Xi; here a subset A⊂ X is open iff A∩ Xiis open in Xifor every i .
The topological space defined in this way is called disjoint union of the topological spaces Xi.
Example 3.11 LetK be a field, n > 0 a given integer and K[x1, . . . , xn] the poly-
nomial ring in n variables and coefficients in K. For any f ∈ K[x1, . . . , xn] we
define
D( f ) = {(a1, . . . , an) ∈ Kn| f (a1, . . . , an) = 0}.
Since D(0) = ∅, D(1) = Knand D( f ) ∩ D(g) = D( f g), the subsets D( f ) form a
basis of open sets for a topology onKnthat is called Zariski topology. One indicates by V( f ) the complement of D( f ), and sets
V(E) = f∈E
V( f ) = {(a1, . . . , an) ∈ Kn| f (a1, . . . , an) = 0 ∀ f ∈ E}
for any subset E ⊂ K[x1, . . . , xn]. The subsets V (E) are, as E varies, the only
closed subsets in the Zariski topology. Given any subset E ⊂ K[x1, . . . , xn] one
writes I = (E) for the ideal generated by E; as E ⊂ I we have V (I ) ⊂ V (E). Conversely, given an element f ∈ I , there is an integer n > 0 and polynomials
f1, . . . , fn ∈ E and g1, . . . , gn∈ K[x1, . . . , xn] such that f = f1g1+ · · · + fngn,
and therefore
V(E) ⊂ V ( f1) ∩ · · · ∩ V ( fn) ⊂ V ( f ) .
This proves that
V(E) ⊂ f∈I
V( f ) = V (I ),
whence V(I ) = V (E) and, further, that Zariski-closed subsets are precisely the subsets of type V(I ), as I varies among the ideals of the ring K[x1, . . . , xn]. Exercises
3.1 (♥) True or false?
1. There is only one topological structure on the singleton.
2. A set with two elements admits exactly four different topological structures. 3. On a finite set any topology has an even number of open sets.
4. On an infinite set with the cofinite topology any pair of non-empty open sets has non-empty intersection.
3.2 Prove that closed intervals[a, b] ⊂ R are closed in the Euclidean topology. 3.3 Let∞ ∈ X be a given element in a set. Check that
3.1 Topological Spaces 43
T = {A ⊂ X | ∞ ∈ A or X − A is finite}
defines a topology on X .
3.4 Let X be a set. Prove that
T = {A ⊂ X | A = ∅ or X − A is finite or countable}
is a topology on X .
3.5 Let(X, ≤) be an ordered set. Show that the subsets
Mx = {y ∈ X | x ≤ y}
are, as x ∈ X varies, a basis for a topology.
3.6 (There exist infinitely many primes,♥) For any pair of integers a, b ∈ Z,
with b> 0, let’s write Na,b= {a + kb | k ∈ Z}. Prove the following facts:
1. arithmetic progressionsB = {Na,b | a, b ∈ Z, b > 0} form a basis for a topology
T on Z;
2. every Na,bis both open and closed inT ;
3. call P= {2, 3, . . .} ⊂ N the set of primes. Then
Z − {−1, 1} = ∪{N0,p| p ∈ P}.
Therefore if P were finite{−1, 1} would be open in T .