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PART I
Basic Definitions
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SECTION 1
General Introduction
A topological space is a pair (X, τ) consisting of a set X and a collection τ of subsets of X, called open sets, satisfying the following axioms:
| O1: | | The union of open sets is an open set. |
| O2: | | The finite intersection of open sets is an open set. |
| O3: | | X and the empty set Ø are open sets. |
The collection τ is called a topology for X. The topological space (X, τ) is sometimes referred to as the space X when it is clear which topology X carries.
If τ1 and τ2 are topologies for a set X, τ1 is said to be coarser (or weaker or smaller) than τ2 if every open set of τ1 is an open set of τ2. τ2 is then said to be finer (or stronger or larger) than τ1, and the relationship is expressed as τ1 ≤ τ2. Of course, as sets of sets, τ1 ⊆ τ2. On a set X, the coarsest topology is the indiscrete topology (Example 4), and the finest topology is the discrete topology (Example 1). The ordering ≤ is only a partial ordering, since two topologies may not be comparable (Example 8.8).
In a topological space (X, τ), we define a subset of X to be closed if its complement is an open set of X, that is, if its complement is an element of τ. The De Morgan laws imply that closed sets, being complements of open sets, have the following properties:
| C1: | | The intersection of closed sets is a closed set. |
| C2: | | The finite union of closed sets is a closed set. |
| C3: | | X and the empty set Ø are both closed. |
It is possible that a subset be both open and closed (Example 1), or that a subset be neither open nor closed (Examples 4 and 28).
An Fσ-set is a set which can be written as the union of a countable collection of closed sets; a Gδ-set is a set which can be written as the intersection of a countable collection of open sets. The complement of every Fσ-set is a Gδ-set and conversely. Since a single set is, trivially, a countable collection of sets, closed sets are Fσ-sets, but not conversely (Example 19). Furthermore, closed sets need not be Gσ-sets (Example 19). By complementation analogous statements hold concerning open sets.
Closely related to the concept of an open set is that of a neighborhood. In a space (X, τ), a neighborhood NA of a set A, where A may be a set consisting of a single point, is any subset of X which contains an open set containing A. (Some authors require that NA itself be open; we call such sets open neighborhoods.) A set which is a neighborhood of each of its points is open since it can be expressed as the union of open sets containing each of its points.
Any collection
of subsets of
X may be used as a
subbasis (or
subbase) to generate a topology for
X. This is done by taking as open sets of
τ all sets which can be formed by the union of finite intersections of sets in
, together with Ø and
X. If the union of subsets in a subbasis
is the set
X and if each point contained in the intersection of two subbasis elements is also contained in a subbasis element contained in the intersection,
is called a
basis (or
base) for
τ. In this case,
τ is the collection of all sets which can be written as a union of elements of
. Finite intersections need not be taken first, since each finite intersection is already a union of elements of
. If two bases (or subbases) generate the same topology, they are said to be
equivalent (Example 28). A
local basis at the point
x ∈
X is a collection of open neighborhoods of
x with the property that every open set containing
x contains some set in the collection.
Given a topological space (X, τ), a topology τY can be defined for any subset Y of X by taking as open sets in τY every set which is the intersection of Y and an open set in τ. The pair (Y,τY) is called a subspace of (X, τ), and τy is called the induced (or relative, or subspace) topology for Y. A set U ⊂ Y is said to have a particular property relative to Y (such as open relative to Y) if U has the property in the subspace (Y,τY) A set Y is said to have a property which has been defined only for topological spaces if it has the property when considered as a subspace. If for a particular property, every subspace has the property whenever a space does, the property is said to be hereditary. If every closed subset when considered as a subspace has a property whenever the space has that property, that property is said to be weakly hereditary.
An important example of a weakly hereditary property is compactness. A space X is said to be compact if from every open cover, that is, a collection of open sets whose union contains X, one can select a finite subcollection whose union also contains X. Every closed subset Y of a compact space is compact, since if {Oα} is an open cover for Y, {Oα} ∪ (X – Y) is an open cover for X. F...
