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

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Discussion Overview

The discussion revolves around the dimensionality of boundaries in n-dimensional spaces, questioning whether the boundary of an n-dimensional space is always n-1 dimensional. The scope includes theoretical considerations and conceptual clarifications regarding boundaries in various dimensional contexts.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants propose that the boundary of an n-dimensional space is indeed n-1 dimensional, using examples such as a 2-dimensional circle having a 1-dimensional boundary.
  • Others argue that the concept of "boundary" is vague and may not apply uniformly across all abstract spaces, citing examples like the graph of the function y = sin(1/x) to illustrate potential ambiguities in defining boundaries.
  • One participant questions the rationale behind using lower-dimensional boundaries to define higher-dimensional spaces, asking why a line cannot effectively divide a 3-dimensional manifold.
  • Another participant asserts that a line does not divide 3-dimensional space into separate parts, suggesting that the boundary concept may not hold as intuitively as proposed.
  • A participant states that a sphere is 2-dimensional and has no boundary, reinforcing the idea that the original question may be meaningless.

Areas of Agreement / Disagreement

Participants express multiple competing views on the nature of boundaries in n-dimensional spaces, with no consensus reached regarding the validity or applicability of the boundary concept across different contexts.

Contextual Notes

Limitations in the discussion include the vagueness of the term "boundary," the dependence on specific definitions, and the presence of pathologies in certain mathematical spaces that complicate the understanding of boundaries.

princeton118
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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 boundary 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 boundary 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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