As part of my research of http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I came across a myriad of different topologies that could be put on various spaces associated with a space . This piece is my attempt to collate and organise them. All are examples of initial topologies. An initial topology is determined by a family of functions.
Initial Topology (Belton 1.24)
Let be a set and
be collection of functions on
, such that
, where
is a topological space, for all
. The initial topology generated by
, denoted by
, is the coarsest topology such that each function
is continuous. It is clear that
is the intersection of all topologies on
that contain
Product Topology (Belton 3.20)
Let be a collection of topological spaces. Their topological product is
, where (
is Cartesian product)
is the Cartesian product of the sets and
is the initial topology generated by the projection maps, for
:
Strong Operator Topology (Belton 2.22)
Let and
be normed spaces; the initial topology on the bounded linear operators
,
generated by the family of maps
(where
is equipped with its norm topology) is called the strong operator topology.
Weak Topology (Belton 3.4)
Any normed space gains a natural topology from its dual space, its weak topology. This is the initial topology generated by
, i.e., the coarsest topology to make each map
continuous. The weak topology on
is denoted by
.
Weak* Topology (Belton 3.19)
Let be a normed vector space. The weak* topology on
is the initial topology generated by the maps
,i.e., the coarsest topology to make these maps continuous. The weak* topology on
is denoted by
.
Leave a comment
Comments feed for this article