Standard Spaces

1. $\mathbb{R}^n = \{(x_1, x_2, \dots, x_n)/ x_i \in \mathbb{R},\; i = 1, 2,\dots,n\}$
2. $\mathbb{C}^n = \{(z_1, z_2, \dots, z_n)/ z_i \in \mathbb{C},\; i = 1, 2,\dots,n\}$
3. $\Vert{\bf x}\Vert = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}$, $\;{\bf x} = (x_1, x_2,\dots, x_n)\in \mathbb{R}^n$
4. $S^{n-1} = \{{\bf x}\in \mathbb{R}^n/ \Vert{\bf x}\Vert = 1\}$
5. $E^n = \{{\bf x}\in \mathbb{R}^n/ \Vert{\bf x}\Vert \leq 1\}$
6. $I^n$ is the standard unit cube, the Cartesian product of $n$ copies of $[0, 1]$.
7. $\dot{I}^n$ is the (topological) boundary of $I^n$.
8. $\mathbb{R}P^n$ is the $n$-dimensional real projective space
9. $M(n,\mathbb{R})$ is the set of all $n\times n$ matrices with real entries
10. $GL(n, \mathbb{R})$ is the set of all invertible $n\times n$ matrices with real entries
11. $O(n, \mathbb{R})$ is the set of all orthogonal matrices with real entries
12. $SO(n, \mathbb{R})$ is the set of all orthogonal matrices with real entries and determinant one
13. $U(n)$ is the set of all $n\times n$ unitary matrices with complex entries
14. $SU(n)$ is the set of all $n\times n$ unitary matrices with determinant one
15. Gr is the category of groups
16. AbGr is the category of abelian groups
17. Top is the category of topological spaces
18. ${\bf Top}^2$ is the category of pairs of topological spaces
19. ${\bf Top}_0$ is the category of pointed topological spaces