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princeton118
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If a space is of n dimension, then the boundary of this space is n-1 dimension or not?
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?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
[tex]y = \sin \left( \frac{1}{x} \right) \quad \quad x \in (0, 1).[/tex]
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 [itex]x = 0 \wedge y \in [-1, 1][/itex])
The boundary of a space is the set of points that define the edge or limit of a given space. It can be thought of as the border or boundary between one space and another.
The boundary of a space is determined by the characteristics and properties of the space itself. It can be defined by geometric features such as edges, surfaces, or volumes, or by other factors such as temperature, pressure, or chemical composition.
The boundary of a space is important in science because it helps us to understand the nature and behavior of different systems and environments. It can also serve as a reference point for studying changes or interactions within a given space.
Yes, the boundary of a space can change over time due to various factors such as physical processes, environmental conditions, or human activities. This can result in alterations or shifts in the characteristics and properties of a space.
The boundary of a space is closely related to other scientific concepts such as boundaries between different states of matter, boundaries between different ecosystems, or boundaries between different levels of organization within a system. It can also be linked to concepts such as interfaces, transitions, or thresholds.