Download Appendix A Point set topology

Appendix A
Point set topology
The concepts from point set topology that are used in this thesis are summarised below. The definitions and results are adapted from Kelley [89], Hocking and Young [79], and Munkres [101].
A topology on a set S is a collection T of subsets of S having the following
properties.
1. The empty set and S are in T .
2. The intersection of each finite subcollection of T is an element of T .
3. The union of each subcollection of T is an element of T .
If T is a topology on S, then (S, T ) is a topological space. If the topology on
S is understood, it is said that S is a topological space.
A subset U of S is defined to be open if U is an element of the topology of
S. A neighbourhood of a point x in a topological space S is a subset of S that
contains an open set containing x.
Consider two topologies T and T 0 on the same set. If T is a subcollection
of T 0 it is said that T 0 is finer than T and T is coarser than T 0 .
A basis for a topology on S is a collection B of subsets of S having the
following properties.
1. For each point x in S, there is an element of B containing x.
2. If x ∈ B1 ∩ B2 for some B1 , B2 ∈ B, then x ∈ B3 ⊆ B1 ∩ B2 for some
B3 ∈ B.
The topology generated by a basis B consists of all finite unions of elements
of B. A subbasis for S is a collection of subsets of S whose union equals S.
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The topology generated by a subbasis S is the collection of all unions of finite
intersections of elements of S. This is the coarsest topology containing S.
A function f : S → T for topological spaces S and T is continuous if for
each open subset V of T , the inverse image f −1 (V ) is open in S. The function
f is continuous at the point x in S if for each open neighbourhood V of f (x),
there is a open neighbourhood U of x such that f (U ) is contained in V . The
function f is continuous if and only if it is continuous at each point in S. A
function f : X → Y is called a homeomorphism if f is a bijection and both f
and f −1 are continuous.
If T is a subset of a topological space S, then the subspace topology for
T consists of all intersections of open sets of S with T . In this topology, it is
said that T is a (topological) subspace of S. If S is a subspace of T , then the
topology on S is called the relative topology induced by T .
Given two topological spaces S and T , the product topology for S × T has
as a basis the collection of all U × V where U is open in S and V is open in T .
Let S be a topological space and let f : S → T be a function. Define Q as
the collection of all subsets Q of T such that f −1 (Q) is open in S. This the
quotient topology for T relative to f and S. It is the finest topology for which
the mapping f is continuous.
A subset C of a topological space S is said to be closed if the complement
S − C is open in S. A point x of S is called a limit point of a subset P of S if
every open neighbourhood of x intersects P in a point other than x. A subset
of S is closed if and only if it contains all its limit points.
The interior of a subset P of S is the union of all open sets of S contained
in P . The closure of a subset P of S is the intersection of all closed sets of
S containing P . A subset P of S is said to be dense in S if Cl (P ) = S. The
boundary of P in S consists of all points that are in the closure of P and not in
the interior of P . A point x in S is in the boundary of P if and only if each open
neighbourhood of x intersects both P and S − P . The interior, the closure and
the boundary of P are denoted by Int (P ), Cl (P ), and Bd (P ), respectively.
A collection S of subsets of S is a cover for S if the union of S equals S.
An open cover consists only of open sets. A space S is compact if every open
cover for S contains a finite subcollection that is also a cover for S. If S is a
subset of Rk , then S is compact if and only if S is closed and bounded.
A topological space S is called a Hausdorff space if for each pair of points
p and q in S, there are disjoint open neighbourhoods U and V , respectively.
Let S and T be topological spaces and let C(S, T ) denote the set of all
continuous functions mapping S into T . The set of functions C(S, T ) is endowed
with a topology as follows. For each compact subset C of S and each open
subset U of T , define the subset S(C, U ) of C(S, T ) by
S(C, U ) = { f | f ∈ C(X, Y ) and f (C) ⊆ U }.
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The set of all such S(C, U ) is a subbasis for the compact-open topology on
C(S, T ). If C(S, T ) has the compact-open topology, then the following function
is continuous:
e : S × C(S, T ) → T : (x, f ) 7→ f (x).
This is called the evaluation map.
Let S be a topological space and (T, d) be a pseudometric space. Let T S
be the set of all functions f : S → T . The topology of compact convergence on
T S is given by the basis defined below. Each basis element is a set of functions
g in T S given by
sup{ d(f (x), g(x)) | x ∈ C },
where f is an element of T S , C is a compact subset of S and > 0. The
compact-open topology on C(S, T ) and the topology of compact convergence
on C(S, T ) (as a subspace of T S ) are the same.
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