13. A little point-set topology.

∗ ∗ ∗ Open and closed sets ∗ ∗ ∗

13.1 Definition. A subset τ of the power set P(X) of a set X is called a topology on X if the following axioms hold: (O1) ∅ ∈ τ and X ∈ τ (O2) S, T ∈ τ ⟹ S ∩ T ∈ τ (O3) S_{i} ∈ τ for all i ∈ I (I an arbitrary index set) ⟹ ⋃_{i∈I} S_{i} ∈ τ (X, τ) is then called a topological space. The members of τ are called open sets.

If (X, τ) is a topological space, then the complements of open sets, A = X \ S with S ∈ τ, are called closed sets. The collection of closed sets C = {X \O | O ∈ τ} satisfies (C1) ∅ ∈ C and X ∈ C. (C2) A, B ∈ C ⟹ A ∪ B ∈ C. (C3) A_{i} ∈ C for all i ∈ I (I an arbitrary index set) ⟹ ⋂_{i∈I} A_{i} ∈ C

Conversely, if C ⊆ P(X) satisfys C1–C3, then the complements of elements of C form a topology on X, whose closed sets are precisely the elements of C.

13.2 Example: Zariski topology on the spectrum of a ring. Let R be a commutative ring. A topology on the spectrum of R (Spec(R) = {P | P a prime ideal of R}) is defined by specifying its closed sets as sets of prime ideals of the form

V (I) = {P ∈ Spec(R) | P ⊇ I},

for some ideal I of R.

There’s nothing to prevent sets from being open and closed at the same time. Sets both open and closed are often called clopen.

13.3 Definition. Let (X, τ) be a topological space.

A collection B of open sets such that every open set is a union of elements of B is called a basis of the topology τ.

A collection S of open sets such that every open set is a union of finite intersections of elements of S is called a subbasis of the topology τ.

Topologies can be defined by specifying a basis or subbasis: If B ⊆ P(X) is closed with respect to finite intersections, then the unions of (arbitrarily many) members of B form a topology on X, of which B is a basis.

If S ⊆ P(X) is any collection of subsets of X, then arbitrary unions of finite intersections of members of S form a topology on X, of which S is a subbasis.

13.4 Example: Order Topology. If (X, ≤) is a totally ordered set, then order topology on X is defined by specifying “open rays”, i.e., sets of the form (a, ∞) = {x ∈ X | a < x} and (−∞, b) = {x ∈ X | x < b}, for a, b ∈ X as a subbasis.

If (X, ≤) has neither a maximal nor a minimal element, then “open intervals” (a, b) = {x ∈ X | a < x < b} form a basis of order topology. (If X does have a maximal or minimal element, then open rays of the form (a, ∞), or (−∞, b), respectively, have to be added to the open intervals to get a basis.)

13.5 Definition. If (X, τ) is a topological space and Y a subset of X then Y inherits a topological structure from X (called subspace topology) through the convention: a subset U of Y is open (in Y ) iff there exists an open subset O of X with U = O ∩ Y .

If Y is an open subset of X then U ⊆ Y is open in Y if and only it is open in X; if Y is a closed subset of X then A ⊆ Y is closed in Y if and only it is closed in X.

13.6 Remark. If (X, ≤) is a totally ordered set and Y ⊆ X, then Y inherits a topology from the order topology of X, and at the same time Y inherits an order relation from X which makes (Y, ≤) a totally ordered set, for which order topology may be defined. These two topologies on Y in general do not agree. (Examples can be found among subsets of the real numbers.)

∗ ∗ ∗ Neighborhoods and neighborhood bases ∗ ∗ ∗

Perhaps a more intuitive approach to topology is through neighborhoods of a point, which (as sets containing an open ball around the point) are already familiar from the study of metric spaces.

13.7 Definition. (∗) Let (X, τ) be a topological space and p ∈ X. A neighborhood of p is a set U such that there exists an open set O with p ∈ O ⊆ U. The set of all neighborhoods of a point p is called the neighborhood filter of p. We will denote in by U(p).

13.8 Remark. Let (X, τ) be a topological space and p ∈ X. The neighborhood filter of p has the properties (U1) ∀U ∈ U(p) p ∈ U. (U2) U, V ∈ U(p) ⟹ U ∩ V ∈ U(p) (U3) U ∈ U(p) and V ⊇ U ⟹ V ∈ U(p) (U4) ∀U ∈ U(p) ∃V ∈ U(p) ∀v ∈ V U ∈ U(v). Also,

(∗)  O ∈ τ ⟺ ∀p ∈ O ∃U ∈ U(p) U ⊆ O

Conversely, given a set X and for each p ∈ X a set U(p) ⊆ P(X) such that U1–U4 holds, we can define a topolgy τ on X by (∗), and, what is more, the neighborhood filter of each point in the resulting topology τ is exactly the U(p) we started out with.

As with metric spaces, it suffices to know a system of “basic” neighborhoods of a point – with the property that every neighborhood contains one of them – to know all neighborhoods.

13.9 Definition. Let (X, τ) be a topological space. A collection B(p) ⊆ U(p) of neighborhoods of p is called a neighborhood basis of p if ∀U ∈ U(p)∃B ∈ B(p) B ⊆ U.

If (X, τ) is a topological space, and for each p ∈ X, B(p) is a neighborhood basis, then (B1) ∀B ∈ B(p) p ∈ B. (B2) U, V ∈ B(p) ⟹ ∃B ∈ B(p) B ⊆ U ∩ V (B3) ∀U ∈ B(p) ∃V ∈ B(p) ∀v ∈ V ∃B ∈ B(v) B ⊆ U).

Also,

(∗∗)  O ∈ τ ⟺ ∀p ∈ O ∃B ∈ B(p) B ⊆ O

Conversely, if we are given for every p ∈ X a collection B(p) of subsets of X satisfying B1–B3, we can define a topology on X by (∗∗), and in this topology B(p) will be a neighborhood basis of p.

13.10 Definition. A topological space in which every point has a countable neighborhood basis is said to satisfy the first countability axiom.

13.11 Example: Let (X, d) be a metric space, and B_{ε}(p) = {x ∈ X | d(p, x) < ε} (for ε > 0, p ∈ X) the open ball of radius ε around p. If we define B(p) as the set of all B_{ε}(p) with ε > 0, then B1–B3 hold. In other words, every metric induces a topology in which the open ε-balls with center p form a neighborhood basis of p. Actually, countably many balls B_{\frac{1}{n}} (p), n ∈ ℕ, already form a neighborhood basis of p in this topology. We see that every metric space satisfies the first countability axiom.

13.12 Example: I-adic topology Let R be a commutative ring and I an ideal of R. I-adic topology on R is defined by specifying a neighborhood base of r ∈ R:

B(r) = {r + I^{n} | n ∈ ℕ}.

Note that these base neighborhoods are both open and closed.

∗ ∗ ∗ Closure and interior ∗ ∗ ∗

13.13 Definition. Let A be a subset of a topological space X. The closure of A, denoted Ā, is defined to be the intersection of all closed sets containing A.

Clearly, Ā is a closed set containing A and every closed set B that contains A also contains Ā.

The closure operator A ↦ Ā on P(X) has the properties: (A1) ¯∅ = ∅. (A2) A ⊆ Ā. (A3) ¯ Ā = Ā. (A4) Ā ∪ B̄ = \overline{A ∪ B}.

Conversely, any operator A ↦ Ā on the power set of a set X with properties A1–A4 may be used to define a topology on X by declaring the closed sets to be precisely the sets of the form Ā for some A ∈ P(X). (To see that these sets satisfy C3, first note that A4 implies A ⊆ C ⟹ Ā ⊆ C̄. From this we get that \overline{⋂_{i∈I} Ā_{i}} ⊆ Ā_{i} for all i ∈ I, i.e., \overline{⋂_{i∈I} Ā_{i}} ⊆ ⋂_{i∈I} Ā_{i} and the reverse inclusion is just A2.)

13.14 Definition. Let (X, τ) be a topological space and A∈P(X) then the interior of A, denoted A^{∘}, is the union of all open sets contained in A.

Clearly, A^{∘} is open and A^{∘} ⊆ A. Also, (X \ A)^{∘} = X \ Ā. (Note that the interior of a non-empty set may be empty.)

∗ ∗ ∗ Boundry points, accumulation points and such ∗ ∗ ∗

The following properties of the closure of a set (which we defined as the intersection of all closed sets containing it) and the interior of a set (which we defined as the union of all open sets contained in it) are often used as definitions:

13.15 Lemma.

• Ā consists of precisely those points p ∈ X such that U ∩ A ≠ ∅ for every neighborhood U ∈ U(p) and

• A^{∘} consists of precisely those points p∈X such that there exists a neighborhood U ∈ U(p) with U ⊆ A.

13.16 Definition. Let (X, τ) be a topological space, p ∈ X and A ⊆ X then

• p is called a boundary point of A if for every neighborhood U ∈ U(p) both A ∩ U ≠ ∅ and (X \ A) ∩ U ≠ ∅. The set of all boundary points of A is the boundary of A, denoted by δA.

• p is called an interior point of A if p ∈ A^{∘}, i.e., if there exists a neighborhood U ∈ U(p) with U ⊆ A.

• p is called an accumulation point of A if every neighborhood of p contains an element of A other than p.

• p is called an isolated point of A if there exists a neighborhood U of p with U ∩ A = {p}.

Note that interior points of A and isolated points of A are necessarily in A, while boundary points and accumulation points may or may not belong to A.

By purely logical arguments we see that every set A induces a partition of X into three disjoint parts in two different ways ( ˙∪ denotes disjoint union):

(i) X = A^{∘} ˙∪ δA ˙∪ (X \ A)^{∘} and (ii) X = {isolated points of A} ˙∪{accumulation points of A} ˙∪(X \ A)^{∘}.

Also, (iii) Ā = A^{∘} ˙∪ δA and (iv) Ā = {isolated points of A} ˙∪ {accumulation points of A} The last two partitions of Ā follow from the characterization of Ā as the set

of those p such that A ∩ U ≠ ∅ for every U ∈ U(p). They are incomparable in the sense that all four combinations of belonging to one set of one partition and one set of the other are possible, an accumulation point of A can be either an interior point or a boundary point of A, etc. Also, we have seen that

(v) Ā = A ∪ δA and (vi) Ā = A ∪ {accumulation points of A} Unlike (iii) and (iv), the unions (v) and (vi) are in general not disjoint.

∗ ∗ ∗ Continuous functions ∗ ∗ ∗

13.17 Definition. Let (X, τ) and (Y, τ′) be topological spaces. A function f: X →Y is called continuous if f^{−1}(O) is open for every open set O ⊆ Y .

In terms of neighborhoods, a function f: X → Y is continuous, if and only if for every x ∈ X, for every U ∈ U(f(x)) there exists a V ∈ U(x) with f(V) ⊆ U. The familiar ε-δ-definition of continuous functions is easily seen to be the specialization to metric spaces of this topological characterization.

13.18 Definition. Let (X, τ) and (Y, τ′) be topological spaces. A function f: X →Y is called open if f(O) is open for every open set O ⊆ X.

A bijective function that is both open and continuous is called a homeomorphism.

A topology τ_{1} on X is called stronger (or finer) than another topology τ_{2} on the same set X if τ_{1} ⊇ τ_{2} (every τ_{2}-open set is τ_{1}-open); τ_{2} is then called weaker or coarser than τ_{1}. Two trivial topologies exist on every set X: discrete topology τ = P(X) (the finest topology on X) and chaotic topology τ = {∅, X} (the coarsest topology on X).

If τ_{1} and τ_{2} are two topologies on a set X then τ_{1} is stronger than τ_{2} iff id_{X} : (X, τ_{1}) → (X, τ_{2}) is continuous; τ_{1} is weaker than τ_{2} iff id_{X} : (X, τ_{1}) → (X, τ_{2}) is open.