Understanding Surface Dimensions for Beginners

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Discussion Overview

The discussion revolves around the definition of the dimension of a surface, exploring the concept of surfaces as two-dimensional entities. Participants seek to clarify the foundational aspects of what constitutes a surface and how its dimensionality is determined.

Discussion Character

  • Exploratory, Conceptual clarification

Main Points Raised

  • One participant questions how to define the dimension of a surface, noting that surfaces are considered 2-D but expressing uncertainty about the reasoning behind this classification.
  • Another participant suggests that dimension can be defined through the concept of homeomorphism, explaining that around any point on a surface, a small enough open subset can be mapped to an open subset of the plane, which is a characteristic of manifolds.
  • A participant raises a question about the definition of "surface," indicating a need for clarity on what qualifies as a surface.
  • In response, another participant offers a non-formal definition, suggesting that a plane or a hollow sphere in R3 could be considered surfaces.

Areas of Agreement / Disagreement

Participants express differing views on the definition of "surface," with some focusing on the dimensionality aspect while others seek clarity on the term itself. The discussion remains unresolved regarding a formal definition of surfaces.

Contextual Notes

Limitations include the lack of formal definitions and the dependence on intuitive understanding of surfaces and dimensions. The discussion does not resolve how to formally define a surface.

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How do we define the dimension of a surface? I know surfaces are 2-D but I don't really get where that comes from.
 
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There are a couple of ways to define dimension. One common one (which explains why surfaces are 2 dimensional) is that around any point if you take a small enough open subset of your surface, there is a homeomorphism (a continuous bijection with a continuous inverse) from that piece of the surface to an open subset of the plane (or R3 or some other power of the reals depending on the dimension of your set). This is the idea of a manifold:

http://en.wikipedia.org/wiki/Manifold

For a surface, you probably have a parametrization which basically describes how to form these homeomorphisms either immediately or with only a little bit of work
 
Perhaps more importantly, how do you define "surface"?
 
HallsofIvy said:
Perhaps more importantly, how do you define "surface"?

I don't really have a formal definition, but I would say something like a plane or a hollow sphere on R3 (ex. x^2+y^2+z^2 = k) is a surface.
 

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