Helly space

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

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:

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

Helly's space is convex as a subset of .

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.

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