I am trying to understand how to do the following two questions concerning compact manifolds:
Show that$\ M\ $is a compact manifold in $\mathbb{R}^{n},\ $then $\partial\ M\ $is also compact; if also $\ M\ $is $\ n\ $-dimensional, then $\partial\ M=\ $bdry$\ M\ $
Show that a compact manifold cannot be represented by a (single) parametric equation.
My confusion is what it means for a manifold to be compact. In some books having to do with advanced calculus or theory of manifolds, it states that the notion of compactness when describing manifolds is distinct from the topological notion of compactness. I also come across questions where it asks to prove properties about compact manifolds without boundary. This just makes it more confusing. It is like saying a closed interval, circle or sphere has no boundary points or an empty boundary, but yet it is closed and bounded. On Wikipedia, compact manifold is discussed in the topic of closed manifolds. Again, I am encountering topological notions associated with compactness. But textbooks says otherwise.
Can someone please help me with some clarifications over my confusions please. Thank you in advance
We know that $M$ is compact manifold in $\mathbb{R}^n$ and the boundary (of any manifold with boundary) $\partial M$ is closed in $M$. Since every closed subset of a compact space is compact, then $\partial M$ is compact. For the problem that any compact manifold cannot be represent as single parametric equation, just note that if we can, then $M$ must be homeomorphic to an open subset of $U \subset \mathbb{R}^{\text{dim }M}$. That is, there exists homeomorphism $\varphi : M\to U = \varphi(M)\subset \mathbb{R}^{\text{dim }M}$. This implies that $U$ is compact (closed and bounded) and also open. Since $U \subset \mathbb{R}^{\text{dim }M}$ is open and closed and $\mathbb{R}^{\text{dim }M}$ connected, this means $U=\mathbb{R}^{\text{dim }M}$. But this is impossible since $U=\varphi(M)$ is compact by continuity.