PlanetMath: thin square

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thin square

(Definition)

Definition 0.1 A tree, is defined here as the underlying space $\vert K\vert$ of a finite

-connected

and boundary $\partial{I}^{2}$ of $I^{2} = I \times I$ (that is, a square (interval) defined here as the Cartesian product of the unit interval

Definition 0.2 A square map $u:I^{2} \longrightarrow X$ in a topological space

is thin if there is a factorisation of

$\displaystyle u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} J_{u} \stackrel{p_{u}}{\longrightarrow} X,$

where $J_{u}$ is a tree and $\Phi_{u}$ is piecewise linear (PWL) on the boundary $\partial{I}^{2}$ of $I^{2}$ .

"thin square" is owned by bci1.

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See Also: copula

Other names:

AlgebraicallyThinSquares

Also defines:

tree, square

Keywords:

thin square, tree, square interval

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Cross-references: thin, topological space, map, real numbers, unit, Cartesian product, interval, boundary, simplicial complex, finite
There are 46 references to this entry.

This is version 16 of thin square, born on 2008-09-04, modified 2008-10-18.
Object id is 10988, canonical name is ThinSquare.
Accessed 454 times total.

Classification:

AMS MSC:	55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
	55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories)
	55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology)
	18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations)

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