Some examples

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

Example 4..1 (The Topology of ${\mathbb{R}}^n$)   We should expect that $X = {\mathbb{R}}^n$ for any integer $n$ is a topological space. This requires characterizing the open sets. An open ball $B(x,\rho)$ is the set of points in the interior of a sphere of radius $\rho$, centered at $x$. Thus,

$$B(x,\rho) = \{x^\prime \in {\mathbb{R}}^n \;\vert\; \Vert x^\prime - x\Vert < \rho\},$$ (4.2)

in which $\Vert\cdot\Vert$ denotes the Euclidean norm (or magnitude) of its argument. The open balls are open sets in ${\mathbb{R}}^n$. Furthermore, all other open sets can be expressed as a countable union of open balls.^4.1 For the case of ${\mathbb{R}}$, this reduces to representing any open set as a union of intervals, which was done so far.

Even though it is possible to express open sets of ${\mathbb{R}}^n$ 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 $H = \{x \in {\mathbb{R}}^n \;\vert\; f(x) \leq 0\}$ 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 ${\cal O}$ that is a closed set. As mentioned in Section 3.1.2, sets constructed only from primitives that use the $<$ relation are open. $\blacksquare$

Example 4..2 (Subspace Topology)   A new topological space can easily be constructed from a subset of a topological space. Let $X$ be a topological space, and let $Y \subset X$ be a subset. The subspace topology on $Y$ is obtained by defining the open sets to be every subset of $Y$ that can be represented as $U \cap Y$ for some open set $U \subseteq X$. Thus, the open sets for $Y$ are almost the same as for $X$, except that the points that do not lie in $Y$ are trimmed away. New subspaces can be constructed by intersecting open sets of ${\mathbb{R}}^n$ with a complicated region defined by semi-algebraic models. This leads to many interesting topological spaces, some of which will appear later in this chapter. $\blacksquare$

Example 4..3 (The Trivial Topology)   For any set $X$, there is always one trivial example of a topological space that can be constructed from it. Declare that $X$ and $\emptyset$ are the only open sets. Note that all of the axioms are satisfied. $\blacksquare$

Example 4..4 (A Strange Topology)   It is important to keep in mind the almost absurd level of generality that is allowed by the definition of a topological space. A topological space can be defined for any set, as long as the declared open sets obey the axioms. Suppose a four-element set is defined as

$$X = \{ , , , \}.$$ (4.3)

In addition to $\emptyset$ and $X$, suppose that $\{$CAT$\}$ and $\{$DOG$\}$ are open sets. Using the axioms, $\{$CAT$,$DOG$\}$ must also be an open set. Closed sets and boundary points can be derived for this topology once the open sets are defined. $\blacksquare$

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 ${\mathbb{R}}^n$ and any configuration spaces that arise in this book, the following is satisfied:

Hausdorff axiom: For any distinct $x_1,x_2 \in X$, there exist open sets $O_1$ and $O_2$ such that $x_1 \in O_1$, $x_2 \in O_2$, and $O_1 \cap O_2 = \emptyset$.

In other words, it is possible to separate $x_1$ and $x_2$ into nonoverlapping open sets. Think about how to do this for ${\mathbb{R}}^n$ by selecting small enough open balls. Any topological space $X$ 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.