III.17 Dimension: Pages 181 – 182.

Dimension is not a single concept.  There are many conflicting possibilities for the definition of dimension.

1.  The number of coordinates.

2.  If a d-1 dimensional closed barrier can be placed between any two points of X then we say X is d-dimensional.  This is the inductive dimensional of a set, Brouwer.

3.  A finite cover of open sets exists which satisfy three coverage properties.

4.  There are homological and cohomological definitions.  The homological definition is the largest d for which some substructure of X has a nontrivial d-th homology group.  This definition can be extended to groups and rings.

5.  The number that corresponds to the best measure of size, such as, length, area, volume and non-integer extensions of these.

The Cantor set is defined by removing middle thirds of the closed interval from zero to one. 

A typographical mistake was found.  1/3 can be written as 0.1 or 0.02222… in ternary expansions.  Not as “0.22222…” which is equal to 1.0.

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