How to qualify a N dimensional manifold as Compact under following condions? Suppose a manifold of N dimensions is closed and bounded in a dimension but it remained unbounded in all other dimensions, so how to categorize the manifold. 
For example, in simpler form how to categorize an infinite strip in a two dimensional euclidean space ?
This infinite strip has a finite thickness in y direction and is unbounded in x direction. 
Is it both open and compact ?  
what kind of properties these kind of these objects can have ? Does it have a boundary if so, what is its boundary ?
 A: The strip
$$X:=\bigl\{(x,y)\in{\mathbb R}^2\>\bigm|\>-1<y<1\bigr\}$$
is an open set in ${\mathbb R}^2$, and is a noncompact two-dimensional manifold. As such it has no boundary: A flat-ant living in $X$ would judge that all neighborhoods look alike, and after some mathematical thinking would come to the conclusion that its world is homeomorphic to ${\mathbb R}^2$, or to a two-dimensional disk.
A boundary is manifesting itself only if you explicitly consider $X$ as a subset of ${\mathbb R}^2$ (the ant knows nothing of this). This means that you are actually considering the space
$$\bar X:=\bigl\{(x,y)\in{\mathbb R}^2\>\bigm|\>-1\leq y\leq 1\bigr\}\ .$$
This space is noncompact as well, but no longer a two-dimensional manifold. It is a two-dimensional manifold with boundary
$$\partial X:=\bigl\{(x,y)\in{\mathbb R}^2\>\bigm||y|=1\bigr\}\ .$$
The neighborhoods of the boundary points are of a different nature from the neighborhoods of the points $x\in X$, and an ant sitting at such a boundary point has definitely a feeling of restriction. (Of course one can describe all of this in an exact manner.)
