Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 space is homeomorphic to a subset of the Hilbert cube.
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Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 space is homeomorphic to a subset of the Hilbert cube.
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Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 space is homeomorphic to a subset of the Hilbert cube.
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In mathematics, the "'Hilbert cube "', named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology.
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An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups and to the fundamental group of the Hilbert cube minus the set
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The group of homeomorphisms of the Hilbert cube [ 0, 1 ] " N " is a universal Polish group, in the sense that every Polish group is isomorphic to a closed subgroup of it.
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If a point in the Hilbert cube is specified by a sequence \ lbrace a _ n \ rbrace with 0 \ leq a _ n \ leq 1 / n, then a homeomorphism to the infinite dimensional unit cube is given by h ( a ) _ n = n \ cdot a _ n.
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Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube \ lbrack 0, 1 \ rbrack ^ \ mathbb { N }, i . e . the countably infinite product of the unit interval ( with its natural subspace topology from the reals ) with itself, endowed with the product topology.
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