Helly space

In mathematics, and particularly functional analysis, the Helly space, named after Eduard Helly, consists of all monotonically increasing functions ƒ : [0,1] → [0,1], where [0,1] denotes the closed interval given by the set of all x such that 0 ≤ x ≤ 1.^[1] In other words, for all 0 ≤ x ≤ 1 we have 0 ≤ ƒ(x) ≤ 1 and also if x ≤ y then ƒ(x) ≤ ƒ(y).

Let the closed interval [0,1] be denoted simply by I. We can form the space I^I by taking the uncountable Cartesian product of closed intervals:^[2]

${\displaystyle I^{I}=\prod _{i\in I}I_{i}}$

The space I^I is exactly the space of functions ƒ : [0,1] → [0,1]. For each point x in [0,1] we assign the point ƒ(x) in I_x = [0,1].^[3]

Helly's space is convex as a subset of ${\displaystyle \mathbb {R} ^{[0,1]}}$.^[4]

Topology

The Helly space is a subset of I^I. The space I^I has its own topology, namely the product topology.^[2] The Helly space has a topology; namely the induced topology as a subset of I^I.^[1] It is normal Haudsdorff, compact, separable, and first-countable but not second-countable.