# Definition of 'simply connected'

In the book 'Lie Groups, Lie Algebras, and Representations' written by Brian C. Hall, a matrix Lie group G is 'simply connected' if it is path-connected and for every continuous path $$A(t),0\le t\le 1$$, lying in G and with $$A(0)=A(1)$$, there exists a continuous function $$A(s,t),0\le s,t\le 1$$, taking values in G and having the following properties: (1) $$A(s,0)=A(s,1)$$ for all s, (2) $$A(0,t)=A(t)$$, and (3) $$A(1,t)=A(1,0)$$ for all t.

But on topology textbooks, it needs to choose a fixed base point, and the loop converges to that base point. While in the previous book, the loop can converge to any point in the loop and a base point is not needed. Are these two definitions equivalent?

• your space is path connected so the choice of base point isn't important since all the fundamental groups are isomorphic. – Bey Alexander Jan 15 at 6:33

## 1 Answer

Yes, they are equivalent under the assumption that the space is path-connected. Of course, in general they are not equivalent.