The definition of a topological space is so general that an incredible variety of topological spaces can be constructed.

(4.2) |

in which denotes the Euclidean norm (or magnitude) of its argument. The open balls are open sets in . Furthermore, all other open sets can be expressed as a countable union of open balls.

Even though it is possible to express open sets of
as unions
of balls, we prefer to use other representations, with the
understanding that one could revert to open balls if necessary. The
primitives of Section 3.1 can be used to generate many
interesting open and closed sets. For example, any algebraic
primitive expressed in the form
produces a closed set. Taking finite unions and intersections of
these primitives will produce more closed sets. Therefore, all of the
models from Sections 3.1.1 and 3.1.2
produce an obstacle region that is a closed set. As mentioned in
Section 3.1.2, sets constructed only from
primitives that use the relation are open.

CAT DOG TREE HOUSE | (4.3) |

In addition to and , suppose that CAT and DOG are open sets. Using the axioms, CATDOG must also be an open set. Closed sets and boundary points can be derived for this topology once the open sets are defined.

After the last example, it seems that topological spaces are so general that not much can be said about them. Most spaces that are considered in topology and analysis satisfy more axioms. For and any configuration spaces that arise in this book, the following is satisfied:

**Hausdorff axiom:** For any distinct
, there exist open sets and such that
,
, and
.

In other words, it is possible to separate and into
nonoverlapping open sets. Think about how to do this for
by
selecting small enough open balls. Any topological space that
satisfies the Hausdorff axiom is referred to as a *Hausdorff
space*. Section 4.1.2 will introduce manifolds, which
happen to be Hausdorff spaces and are general enough to capture the
vast majority of configuration spaces that arise. We will have no
need in this book to consider topological spaces that are not
Hausdorff spaces.

Steven M LaValle 2012-04-20