关于不同维度空间区域对应术语的技术问询

提问：What is the term for a three dimensional region of space - as we have surface and curve for two and one dimensions respectively? I've sometimes seen this called a volume, although I find that misleading, as volume is typically a measure (a real number) on such a region. Likewise for higher dimensions.

Great call pointing out this ambiguity—everyday language often blurs the line between a geometric object and the measure used to describe it, so your confusion is totally understandable!

For a 3D spatial region (the direct analog to curves in 1D and surfaces in 2D), the precise mathematical term for the region itself is usually a solid (or more formally, a 3-manifold with boundary if we’re talking about a smooth, well-behaved region). You’re absolutely right that "volume" is misleading here—mathematically, volume refers specifically to the scalar measure of how much space that 3D region occupies, not the region itself.

When it comes to higher dimensions (n > 3), the general term we use is an n-dimensional manifold with boundary (or just n-manifold if the region has no defined boundary). In more casual or applied contexts, people might also refer to it simply as an "n-dimensional region". For simpler, structured shapes, you might encounter terms like n-cell (a generalized version of a line segment, square, cube, etc.) in topology, but that’s reserved for specific, uniform regions.

The core takeaway here is keeping this key distinction clear:

• Geometric object: The physical region/shape (curve, surface, solid, n-manifold)
• Measure: The numerical value describing its size (length, area, volume, n-volume)

备注：内容来源于stack exchange，提问作者SRobertJames