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While enjoying Lie groups with Brian C. Hall's "Lie groups, Lie algebras, and representations", I'm stuck with the "standard argument using the compactness of the interval $[0,1]$" in the proof of the the Corollary 2.31 which I repeat here:

Corollary 2.29] If G is a matrix Lie group with Lie algebra $\frak{g}$, there exists a neighborhood U of 0 in $\frak{g}$ and a neighborhood V of I in G such that the exponential mapping takes U homeomorphically onto V.

Corollary 2.31] If G is a connected matrix Lie group, then every element A of G can be written in the form $$A=e^{X_1}e^{X_2}\cdots e^{X_m}$$ for some $X_1, X_2,\dots,X_m$ in $\frak{g}$.

Proof of Corollary 2.31] Since G is connected, we can find a continuous path $A(t)$ in G with $A(0)=I$ and $A(1)=A$. Let V be a neighborhood of I in G as in Corollary 2.29, so that every element of V is the exponential of an element of $\frak{g}$. A standard argument using the compactness of the interval $[0,1]$ shows that we can pick a sequence of numbers $t_0,\dots,t_m$ with $0=t_0<t_1<\cdots<t_m=1$ such that $$A^{-1}_{t_{k-1}}A_{t_k}\in V$$ for all $k=1,\dots,m$. Then, $$A=(A^{-1}_{t_{0}}A_{t_1})(A^{-1}_{t_{1}}A_{t_2})\cdots (A^{-1}_{t_{m-1}}A_{t_m}).$$ If we choose $X_k\in \frak{g}$ with $\exp X_k=A^{-1}_{t_{k-1}}A_{t_k}$ $(k=1,\dots,m)$, we have $$A=e^{X_1}e^{X_2}\cdots e^{X_m}.$$

Although I could guess what geometric picture he wants to convey, can somebody elucidate what the "standard argument using the compactness of the interval $[0,1]$" is and how it was used to obtain $A^{-1}_{t_{k-1}}A_{t_k}\in V$ (for a physicist who knows the basic concept of compactness but unfamiliar with the usual technique to utilize it)?

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Given $t\in[0,1]$ there exists an open nhbhd $U_t$ of $t$ in $[0,1]$ such that for every $s\in U_t$ $A_s^{-1}A_t\in V$ (since $A_t^{-1}A_t=I\in V$ and $V$ is open). The collection of $U_t$'s give you an open cover of $[0,1]$. The "standard compactness argument" referred above is that since $[0,1]$ is compact you can select a finite subcover of $[0,1]$, $U_{s_0},\ldots, U_{s_p}$ and if you arrange indexes so that $s_0<s_1<\ldots <s_p$ and take $U_{s_i}=[t_i,t_i+1]$ you have what you're aiming at.

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  • $\begingroup$ Or use the existence of Lebesgue's number for the covering. $\endgroup$ – Mariano Suárez-Álvarez May 4 '15 at 9:53
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    $\begingroup$ Yes, but not when talking to "a physicist who knows the basic concept of compactness but unfamiliar with the usual technique to utilize it" ;D $\endgroup$ – N. Ciccoli May 4 '15 at 9:57
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    $\begingroup$ Bit you are talking to someone who tells you up front that he is enjoying a textbook ok Lie groups! Sieze the opportunity. $\endgroup$ – Mariano Suárez-Álvarez May 4 '15 at 17:30
  • $\begingroup$ @N.Ciccoli I guess that your concise answer (implicitly) uses a continuous function $\phi_t:[0,1]\to G$ for $t\in [0,1]$ to construct an open cover of $[0,1]$ by $U_t=\phi^{-1}(V)$, and finally a finite cover through the compactness of $[0,1]$. Thanks! $\endgroup$ – eneron May 6 '15 at 8:27
  • $\begingroup$ The continuous function $A(t)$ is the same thing as $\phi_t$, isn't it? $\endgroup$ – N. Ciccoli May 6 '15 at 8:30

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