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. 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 II by taking the uncountable Cartesian product of closed intervals:
 * $$ I^I = \prod_{i \in I} I_i $$

The space II 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 Ix = [0,1].

Topology
The Helly space is a subset of II. The space II has its own topology, namely the product topology. The Helly space has a topology; namely the induced topology as a subset of II. It is normal Haudsdorff, compact, separable, and first-countable but not second-countable.