Topological name for $D^m \times \mathbb{R}^n$

Question

Is there a name for a topological space homeomorphic to $D^m \times \mathbb{R}^n$, where $D^m$ is a closed $m$-dimensional ball?

I would call it $(m,n)$-cylinder if there is no other conventional name. But maybe there is?

Answer 1

The word "cylinder" is quite overloaded in mathematics, and does not have an entirely unambiguous meaning. For example:

• In elementary geometry (i.e. Euclidean geometry, as might be taught to elementary or high school students), a cylinder is the Cartesian product of a disk and a closed interval—it is a solid object with a top and bottom, with horizontal cross-sections which are disks.

• In general topological contexts, a cylinder (over a space $X$) is generally defined to be a space which is the Cartesian product $X$ and an interval (either $[0,1]$ if one wants to consider a "finite" (or, perhaps, closed or compact) cylinder, or $\mathbb{R}$ if one wants to consider an unbounded (or, maybe, open) cylinder).

• In mathematical physics / functional analysis, there are objects called "cylinder sets" (not quite cylinders, but, perhaps, related) which are, more or less, collections of functions of time ("paths", not necessarily continuous) which pass through specified Borel sets at particular moments in time.

Thus any time one uses the word "cylinder" in a mathematical context, it is likely advisable to be specific about what, precisely, is meant.

That being said, I have never encountered sets or spaces of the form $D^m \times \mathbb{R}^n$ as interesting objects of study in-and-of themselves. While I work in a narrow field of mathematics (fractal geometry) where such objects aren't likely to come up, and therefore cannot speak for all mathematicians, I suspect that these objects are not common or interesting enough to a broad range of mathematicians to have earned a particular name. Assuming that you define the term "$(m,n)$-cylinder" near the beginning of any document in which you use the term, I doubt that you will have any trouble or cause any controversy. Personally, I think that the following definition is entirely reasonable:

Definition: An $(m,n)$-cylinder is a set of the form $D^m\times\mathbb{R}^n$, where $m$ and $n$ are integers, and $D^m$ is a closed ball in $\mathbb{R}^m$.