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 .

### Like this:

Like Loading...

*Related*

## Leave a comment

Comments feed for this article