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 II by taking the uncountable Cartesian product of closed intervals:[2]
- [math]\displaystyle{ I^I = \prod_{i \in I} I_i }[/math]
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].[3]
Topology
The Helly space is a subset of II. The space II has its own topology, namely the product topology.[2] The Helly space has a topology; namely the induced topology as a subset of II.[1] It is normal Haudsdorff, compact, separable, and first-countable but not second-countable.
References
- ↑ 1.0 1.1 Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, pp. 127 − 128, ISBN 0-486-68735-X
- ↑ 2.0 2.1 Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, p. 125 − 126, ISBN 0-486-68735-X
- ↑ Penrose, R (2005). The Road to Reality: A Complete guide to the Laws of the Universe. Vintage Books. pp. 368 − 369. ISBN 0-09-944068-7.
Gelfand–Shilov space
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