Let follow this definition of manifold. An n-manifold is a Hausdorff Topological space, Such That Each point you have an open neighborhood homeomorphic to the open disc $ U^n = \left\{ {x \in R^n :\left| x \right| < 1} \right\} $
Let this set:
$ X = \left\{ {\left( {x,y} \right) \in R^2 /\,\,x \in \left[ { - 10,10} \right],\,y \in \left[ { - 1,1} \right]} \right\} $
Define the quotient (10, y) related to (-10,-y) for -1
In the book ( Massey) comes out that if we consider the edges $ y=1 ,y=-1 $ Would not be a manifold under our definition. I do not understand why anyway would say that a manifold with boundary, if it refers not to ask more things or completely different. I'm just learning this quotient topology and I can not imagine the folds and stuff) =. If someone can give me advice on what to do with that, I really appreciate it
Edit: (TB) In order to give some context, let me quote the relevant passage from Massey, (assuming A basic course in algebraic topology, Springer GTM 127 was meant).
From the bottom of page 3:
The simplest example of a $2$-dimensional manifold exhibiting this phenomenon [non-orientability] is the well-known Möbius strip. As the reader probably knows, we construct a model of a Möbius strip by taking a long, narrow rectangular strip of paper and gluing the ends together with a half twist (see Figure 1.1). Mathematically, a Möbius strip is a topological space that is described as follows. Let $X$ denote the following rectangle in the plane: $X = \{(x,y) \in \mathbf{R}^2 : -10 \leqq x \leqq +10, \;-1 \lt y \lt +1\}.$ We then form a quotient space of $X$ by identifying the points $(10,y)$ and $(-10,-y)$ for $-1 \lt y \lt +1$. Note that the two boundaries of the rectangle corresponding to $y=+1$ and $y=-1$ were omitted. This omission is crucial; otherwise the result would not be a manifold (it would be a “manifold with boundary,” a concept we will take up in Chapter XIV [more precisely, XIV.§7, p.375ff]). Alternatively, we could specify a certain subset of $\mathbf{R}^3$ which is homeomorphic to the quotient space just described.
Unfortunately, Google managed to garble Figure 1.1, so here it is in full: