diff options
Diffstat (limited to 'ts')
| -rw-r--r-- | ts/n4.lyx | 46 |
1 files changed, 41 insertions, 5 deletions
@@ -1903,16 +1903,52 @@ Si status open \begin_layout Plain Layout -\begin_inset Note Note -status open +Si +\begin_inset Formula $f$ +\end_inset -\begin_layout Plain Layout -Demostración. -\end_layout + no es sobreyectiva, existe +\begin_inset Formula $z_{0}\in\mathbb{S}^{1}$ +\end_inset + + tal que +\begin_inset Formula $f(\mathbb{S}^{1})\subseteq\mathbb{S}^{1}\setminus\{z_{0}\}\cong(0,1)$ +\end_inset + + con el homomorfismo +\begin_inset Formula $h:(0,1)\to\mathbb{S}^{1}\setminus\{z_{0}\}$ +\end_inset + + dado por +\begin_inset Formula $h(t):=e(t+\theta_{0})$ +\end_inset +, donde +\begin_inset Formula $e(\theta_{0}):=z_{0}$ +\end_inset + +. + Por tanto, si +\begin_inset Formula $\alpha_{f}(s):=f(e(s))$ +\end_inset + +, existe un levantamiento de +\begin_inset Formula $\alpha_{f}$ \end_inset + dado por +\begin_inset Formula $\tilde{\alpha}_{f}(s):=\theta_{0}+h^{-1}(\alpha_{f}(s))$ +\end_inset + +, pues +\begin_inset Formula $e(\tilde{\alpha}_{f}(s))=e(\theta_{0}+e|_{\mathbb{S}^{1}\setminus\{z_{0}\}}^{-1}(\alpha_{f}(s))-\theta_{0})=\alpha_{f}(s)$ +\end_inset +, pero entonces +\begin_inset Formula $\tilde{\alpha}_{f}(1)-\tilde{\alpha}_{f}(0)=h^{-1}(\alpha_{f}(1))-h^{-1}(\alpha_{f}(0))\overset{\alpha_{f}(1)=\alpha_{f}(0)}{=}0$ +\end_inset + +. \end_layout \end_inset |