Helly space

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

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:

$I^I=\prod _{i\in I}{I}_i$

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

Topology

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

References

Gelfand–Shilov space