John Stillwell - Naive Lie Theory.pdf - Index of

164 8 Topology
8.2 Closed matrix groups
In Lie theory, closed sets are important from the beginning, because all
matrix Lie groups are closed sets in the appropriate topology. This has to do
with the continuity of matrix multiplication and the determinant function,
which we assume for now. In the next section we will discuss continuity
and its relationship with open and closed sets more thoroughly.
Example 1. The circle group S 1 = SO(2).
Viewed as a set of points in C or R 2 , the unit circle is a closed set
because its complement (the set of points not on the circle) is clearly open.
Figure 8.1 shows a typical point P not on the circle and an ε-neighborhood
of P that lies in the complement of the circle. The open neighborhood of P
is colored gray and its perimeter is drawn dotted to indicate that boundary
points are not included.
P
Figure 8.1: Why the complement of the circle is open.
Example 2. The groups O(n) and SO(n).
We view O(n) as a subset of the space R n2 of n×n real matrices, which
we also call M n (R). The complement of O(n) is
M n (R) − O(n)={A ∈ M n (R) : AA T ≠ 1}.
This set is open because if A is a matrix in M n (R) with AA T ≠ 1 then
some entries of AA T are unequal to the corresponding entries (1 or 0) in 1.
It follows, since matrix multiplication and transpose are continuous, that
BB T also has entries unequal to the corresponding entries of 1 for any B
sufficiently close to A. Thus some ε-neighborhood of A is contained in
M n (R) − O(n),soO(n) is closed.