Understanding Switzer's telescope construction on a CW complex It is my understanding that, given some filtration $(X^n)_{n \geq 0}$ of a space $X$, or more generally a sequence $X^0 \to X^1 \to X^2 \to \cdots$ of spaces and maps, one usually constructs the "telescope" of the sequence as the subspace $\bigcup_{n \geq 0} [n,n+1] \times X^n \subset [0,\infty) \times X$ (or, in the general case, as a sequence of mapping cylinders glued together at their ends). I am trying to relate this to the following construction in Switzer's Algebraic Topology: 


I am confused by the presence of smash products and the added disjoint points on the intervals (the notation $A^+$ means $A$ with an added disjoint point), and also the fact that the basepoints are not explicitly mentioned. Is it to be understood that the basepoint of $[n-1,n]^+$ is the added point, or some endpoint of the interval? Either possibility yields a distinct smash product with the $n$-skeleton $X^n$, but in any case I don't see how the end result is supposed to look remotely like the usual telescope. What is the intuition behind this construction? Why smash the $n$-skeleton with an interval with an added disjoint point? Any insight would be greatly appreciated.
 A: The notation $X^+$ always considers the disjointly added point as the marked point. Switzer's construction is homotopy equivalent to the one that you describe, the difference is in the formal category-theoretical properties. Algebraic topology is generally concerned with pointed spaces: you can't define homotopy groups otherwise, the cohomology are also better behaved. The universal way to convert a space to a pointed space is precisely to add a disjoint base point. Formally $X^+$ is the left adjoint functor to the forgetful functor from pointed spaces to spaces. The telescope can be formally defined as the homotopy colimit of the sequence in question. A homotopy colimit in topology can be explicitly constructed as a usual colimit aka glueing of spaces multiplied by the intervals. In the nonpointed category this reduces to the construction that you described, in the pointed category it reduces to Switzer's one. As a more elemetary motivation, note that you need the telescope to be equipped with a base point and it must be compatible with the base points of $X^i$. The simplest way to achieve that is to collapse all subspaces of the form $[n,n+1]\times x_i,\ x_i \in X^i$ into a single point. This is exactly what Switzer's telescope does.
