In
mathematics, the unit interval is the
interval [0,1], that is the set of all
real numbers x such that zero is less than or equal to x and x is less than or equal to one. The unit interval plays a fundamental role in
homotopy theory, a major branch of
topology. It is a
metric space,
compact,
contractible,
path connected and
locally path connected. As a
topological space, it is
homeomorphic to the
extended real number line. The unit interval is a
one-dimensional analytical
manifold with boundary {0,1}, carrying a standard
orientation from 0 to 1. As a subset of the real numbers, its
Lebesgue measure is 1. It is a
totally ordered set and a
complete lattice (every subset of the unit interval has a
supremum and an
infimum).
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