Configuration space of ordered points in the plane

Definition
Fix a natural number $$n$$ This topological space is defined as the configuration space of $$n$$ ordered points in the Euclidean plane $$\R^2$$, which can also be thought of as the space of complex numbers $$\mathbb{C}$$.

It is a classifying space for the pure braid group of degree $$n$$. In particular, the homology groups of this space are the homology groups of the pure braid group of degree $$n$$, viewed as a group.

Related notions

 * Configuration space of unordered points in the plane: This can also be thought of as the classifying space for the braid group.