4b: What is topological dimension?
Description
This article is from the Fractal FAQ, by Ermel Stepp stepp@muvms6.mu.wvnet.edu with numerous contributions by others.
4b: What is topological dimension?
Topological dimension is the "normal" idea of dimension; a point has
topological dimension 0, a line has topological dimension 1, a surface has
topological dimension 2, etc.
For a rigorous definition:
A set has topological dimension 0 if every point has arbitrarily small
neighborhoods whose boundaries do not intersect the set.
A set S has topological dimension k if each point in S has arbitrarily small
neighborhoods whose boundaries meet S in a set of dimension k-1, and k is the
least nonnegative integer for which this holds.
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