Editing Hilbert cube (section)

==Properties==

As a product of [[compact (topology)|compact]] [[Hausdorff space]]s, the Hilbert cube is itself a compact Hausdorff space as a result of the [[Tychonoff theorem]].
The compactness of the Hilbert cube can also be proved without the [[axiom of choice]] by constructing a continuous function from the usual [[Cantor set]] onto the Hilbert cube.

In <math>\ell_2,</math> no point has a compact [[neighbourhood (topology)|neighbourhood]] (thus, <math>\ell_2</math> is not [[locally compact]]). One might expect that all of the compact subsets of <math>\ell_2</math> are finite-dimensional. The Hilbert cube shows that this is not the case. But the Hilbert cube fails to be a neighbourhood of any point <math>p</math> because its side becomes smaller and smaller in each dimension, so that an [[open ball]] around <math>p</math> of any fixed radius <math>e > 0</math> must go outside the cube in some dimension.

The Hilbert cube is a convex set, whose span is dense in the whole space, but whose interior is empty. This situation is impossible in finite dimensions. The closed tangent cone to the cube at the zero vector is the whole space.

Let <math>K</math> be any infinite-dimensional, compact, convex subset of <math>\ell_2</math>; or more generally, any such subset of a [[locally convex topological vector space]] such that <math>K</math> is also metrizable; or more generally still, any such subset of a metrizable space such that <math>K</math> is also an [[Retraction (topology)#Absolute neighborhood retract (ANR)|absolute retract]]. Then <math>K</math> is homeomorphic to the Hilbert cube. {{sfnp|Sakai|2020|p=x}}

Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore [[Normal space|T4]]) and [[second countable]]. It is more interesting that the converse also holds: Every [[second countable]] [[Normal space|T4]] space is homeomorphic to a subset of the Hilbert cube.

In particular, every G<sub>δ</sub>-subset of the Hilbert cube is a [[Polish space]], a topological space homeomorphic to a separable and complete metric space. Conversely, every Polish space is homeomorphic to a [[Gδ set|G<sub>δ</sub>-subset]] of the Hilbert cube.{{sfn|Srivastava|1998|p=55}}