Is the Boundary of an n-Dimensional Space Always n-1 Dimensional?

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In n-dimensional space, the boundary is typically considered to be n-1 dimensional, as illustrated by examples like a 2D circle having a 1D boundary and a sphere being a 2D surface without a boundary. However, the concept of "boundary" can be ambiguous and problematic in abstract spaces, leading to confusion and pathologies. For instance, defining the boundary of certain curves, such as y = sin(1/x) for x in (0, 1), raises questions about dimensionality and closure. Additionally, a line cannot effectively divide a 3D manifold, as it allows for continuous paths connecting points without crossing the line. Ultimately, the discussion highlights the complexities and limitations of defining boundaries in higher-dimensional spaces.
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If a space is of n dimension, then the boundary of this space is n-1 dimension or not?
 
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If by the edge, then that is correct. If you have a 2 dimensional circle, the outer rim or boundry is a curved one dimensional line. If you have a sphere, the outer edge is a cirved two dimensional hollow sphere.

Basically, the boundary of a space of dimension n is n-1, however, the curving of the boundry is in n space.

I hope that was what you were asking.
 
Thanks
 
Alas, his question was incredibly vague; as stated it doesn't make any sense, because the concept of "boundary" doesn't really make sense for an abstract space, and there are lots of pathologies even for "usual" spaces.

For example, consider the graph of the function

y = \sin \left( \frac{1}{x} \right) \quad \quad x \in (0, 1).

How are you going to define the boundary of this curve? Once you've chosen a definition, is it zero-dimensional? (Note that the closure of the graph of this curve consists of the entire line segment x = 0 \wedge y \in [-1, 1])
 
Hurkyl said:
Alas, his question was incredibly vague; as stated it doesn't make any sense, because the concept of "boundary" doesn't really make sense for an abstract space, and there are lots of pathologies even for "usual" spaces.

For example, consider the graph of the function

y = \sin \left( \frac{1}{x} \right) \quad \quad x \in (0, 1).

How are you going to define the boundary of this curve? Once you've chosen a definition, is it zero-dimensional? (Note that the closure of the graph of this curve consists of the entire line segment x = 0 \wedge y \in [-1, 1])
Say it more clearly, why we use a line or curve to divide the 2 dimension manifold, why we use a 2 dimension surface to divide the 3 dimension manifold?
Why we can't use a line to divide the 3 dimension manifold?
 
Because it doesn't divide it! If you draw a line in 3 dimensions, you can draw a smooth curve from any point, not on the circle, to any other point, not on the circle, without crossing the line. A line does NOT divide 3 dimensional space into two separate parts.
 
A sphere is 2-dimensional. It has no boundary. The question is, as pointed out by Halls, meaningless.
 

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