Just some random challenges about geometry and topology.
Prove it or than post some others: there are no simple questions, just elegant solutions!
Please use LaTeX to post answers and other problems!

First (easy) Challenge: Let $\mathcal{M}[\math] be a smooth connected manifold. Let <script type="math/tex">\\mathcal{N} \subset \mathcal{M}[\math] be a smooth, closed and connected hypersurface. Prove that the smooth manifold <script type="math/tex">\\mathcal{M} \setminus \mathcal{N}[\math] has either one or two connected components.$