Download Basic Notions Of Topology

Basic Notions Of Topology
Topological Spaces, Bases and Subbases, Induced Topologies
Let X be an arbitrary set. A system O of subsets of X is called a topology on X, if the following
holds:
a) The union of every class of sets in O is a set in O, i.e. for an arbitrary index set I we
have:
[
Oi ∈ O for all i ∈ I =⇒
Oi ∈ O.
i∈I
b) The intersection of every finite class of sets in O is a set in O, i.e. for all n ∈ N we have:
O1 , . . . , On ∈ O =⇒
n
\
Oi ∈ O.
i=1
c) X, ∅ ∈ O.
A topological space is a pair (X, O) consisting of a set X and a topology O on X. The sets in
O are called the open sets; the complement of an open subset of X is called closed.
A system B of open subsets of a topological space (X, O) ist called an open basis of the topology,
if every open set O ∈ O is the union of sets from B, i.e. for all O ∈ O holds:
∀ x ∈ O ∃ B ∈ B : B ⊂ O and x ∈ B.
An open subbasis of (X, O) is a system S ⊂ O such that the set of all finite intersections of
sets from S forms an open basis of O. An open bases and respectively an open subbasis is
called countable, if it only consists of countably many sets.
Let X be an arbitrary set and S any class of subsets of X. Then there exists only one topology
O on X for which S forms a subbases. This topology simply consists of the unions of finite
intersections of sets in S.
Example: Let F be class of all closed subsets of Rn . We equip F with the unique topology for
which the system
{F C : C ∈ C} ∪ {FG : G ∈ G}
forms an open subbasis. This topology is usually called the topology of closed convergence. The
system
τ := {FGC1 ,...,Gk : C ∈ C, G1 , . . . , Gk ∈ G, k ∈ N0 },
where FGC1 ,...,Gk := F C ∩ FG1 ∩ · · · ∩ FGk , is an open basis of the topology of closed convergence.
Furthermore there exists a subset τ 0 ⊂ τ such that τ 0 is a countable open basis of the topology
of closed convergence.
If U ⊂ X is a subset of a topological space (X, O), then the class OU := {O ∩ U | O ∈ O} of
subsets of U forms a topology on U , the so-called induced topology. (U, OU ) is usually referred
to as a subspace of X.
Example: C ⊂ F can be equipped with the topology induced by the topology of closed
convergence.
If O and O0 both are topologies on the same set X, we say that O is finer than O0 , if O0 ⊂ O.
O0 on the other hand is then said to be coarser than O.
Example: On C the topology generated by the Hausdorff metric is finer than the topology
induced by the topology of closed convergence.
Neighbourhoods, Convergence, Haussdorff Spaces
Let (X, O) be a topological space and x ∈ X. A subset N ⊂ X is called a neighbourhood of x,
if there exists an open set O ∈ O such that O ⊂ N and x ∈ O.
A sequence (xn )n∈N in a topological space (X, O) is said to converge to x ∈ X, if for every
neighbourhood U of x there exists a n0 ∈ N such that xn ∈ U for all n ≥ n0 .
Example: Let (Fj )j∈N be a sequence in F and F ∈ F. Then the following statements a)and b)
are equivalent:
a) Fj → F for j → ∞ in the topology of closed convergence.
b) Both c1) and c2) hold:
c1) For all x ∈ F there exists for almost all j ∈ N an element xj ∈ Fj such that xj → x
for j → ∞.
c2) For every subsequence (Fjk )k∈N and every convergent sequence (xjk )k∈N with xjk ∈
Fjk we have limk→∞ xjk ∈ F .
Let (X, O) have a countable basis. Then the following are equivalent:
a) A ⊂ X is closed.
b) For every sequence (an )n∈N with an → a we have a ∈ A.
A Hausdorff space is a topological space in which each pair of distinct points can be separated
by disjoint neighbourhoods.
Example: F equipped with the topology of closed convergence is a Hausdorff space.
Continuous Mappings
A mapping f : X → Y between two topological spaces (X, O1 ) and (Y, O2 ) is called continuous
if the preimage of every open set in Y is an open set in X, i.e.
f : X → Y continuous
:⇐⇒
∀ O ∈ O2 : f −1 (O) ∈ O1 .
Let S2 be an open subbasis of O2 . Then f is continuous if and only if
∀S ∈ S2 : f −1 (S) ∈ O1 .
Let (X, O) have a countable basis. Then f : X → Y is continuous if and only if for all x ∈ X
holds:
For all (xn )n∈N with xn → x we have f (xn ) → f (x).
Compactness
A topological space (X, O) is called compact, if every open cover of X has a finite subcover, i.e.
for every index set I we have:
[
[
Oi = X and Oi ∈ O for all i ∈ I =⇒ ∃ I 0 ⊂ I :| I 0 |< ∞ and
Oi = X.
i∈I 0
i∈I
We call a subset A ⊂ X of X compact, if the subspace (A, OA ) is a compact.
(X, O) is said to be locally compact, if it is a Hausdorff space and for each element x ∈ X there
is a compact neighbourhood of x.
Every compact Hausdorff space (X, O) with a countable basis is sequential compact, i.e. every
sequence (xn )n∈N in X has a convergent subsequence.
Example: F is compact, F 0 := F \ {∅} is locally compact but not compact.
Metric Spaces and Metrizability
A set X is called a metric space if there exists a mapping d : X × X → [0, ∞) with the following
properties:
a) d(x, y) = 0
⇐⇒
x = y,
b) d(x, y) = d(y, x) for all x, y ∈ X,
c) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.
We say that a subset O ⊂ X is open if for every x ∈ X there is an r > 0 such that
B(x, r) := {y ∈ X | d(x, y) < r} ⊂ O.
The class O := {O ⊂ X | O open } of all open subsets of X forms a topology on X, usually
called the topology generated by d.
Example: C equipped with the Hausdorff metric d.
Let X be a set equipped with two metrics d and d0 . Let O and O0 denote the topologies
generated by d and d0 respectively. Then O is finer than O0 if and only if every sequence
(xn )n∈N in X that converges to x ∈ X with respect to d converges to x with respect to d0 as well.
A metrizable space is a topological space (X, O) with the property that there exists a metric d
on the set X whose generated topology is precisely O.
Both compact Hausdorff spaces and locally compact spaces are metrizable if and only if they
have a countable bases.
Example: F and F 0 are metrizable.