TITLE: Does the torus have a boundary? And about the concept of boundary.
QUESTION [6 upvotes]: I am getting confused with the concept of boundary. So I would like to see what a boundary is by using examples. So does the torus have a boundary?

REPLY [3 votes]: As illustrated by the above answers, there are several meaning of the term boundary, and the precise notion of the boundary of a subset of a topological space is one of them. 
Another is of course the notion of the boundary of an $n $-manifold with boundary. So if we consider a  solid disk $D $ in the plane it is a $2 $-manifold with boundary a circle $C$, and it's boundary as a subset of the plane is also the same circle. 
The notion of "boundary" has a more confusing history in algebraic topology, where the early writers (e.g. Betti) wanted to talk about "cycles modulo boundaries" but the notions were not clear. Poincaré's famous papers on "Analysis situs" (starting in 1895) cleared up many things by using simplicial decompositions and the notion of $n $-chain as a formal sum of oriented $n $-simplices, so that the boundary of an $n$-simplex had a well defined meaning. This was further clarified by Eilenberg (1944) with the notion of an ordering on the vertices of a simplex, so the that the $i$'th face $\partial_i  \sigma$ was well defined and we could then define $$\partial \sigma = \sum _i (-1)^i \partial _i \sigma$$
leading to $\partial \partial=0 $. Hence a cycle was a chain $z $ with $\partial z=0 \,$ and a boundary $b $ was a chain such that $b=\partial c $ for some $c $. Then every boundary was a cycle, as required for the notion of homology. 
There is more to be said on this story. But the point I wanted to make is that the notion of "boundary" has a long history and it is not unreasonable to be confused on the intuition. 
Nov 19, 2016 See this paper by M. Barr on the relations between oriented and ordered singular theories.