Tikhonov cube

The topological product of $\tau$ copies of the unit interval $I$ of the real line, where $\tau$ is an arbitrary cardinal number. It is denoted by $I^\tau$. The Tikhonov cube was introduced by A.N. Tikhonov in 1929. If $\tau=n$ is a natural number, then the Tikhonov cube $I^\tau$ is the unit cube in $n$-dimensional Euclidean space, and its topology is induced from the scalar-product metric. If $\tau=\aleph_0$ is the cardinality of the natural numbers, then the cube $I^\tau$ is homeomorphic to the Hilbert cube. For $\tau_1\neq\tau_2$, the Tikhonov cubes $I^{\tau_1}$ and $I^{\tau_2}$ are not homeomorphic: If $\tau$ is an infinite cardinal number, then $\tau$ is the weight of $I^\tau$ (cf. Weight of a topological space), while if $\tau=n$ is a natural number, then $n$ is the dimension of $I^\tau$. Two properties of the Tikhonov cubes $I^\tau$ are particularly important: the compactness of each of them, independently of $\tau$, and their universality with respect to completely-regular $T_1$-spaces of weight not exceeding $\tau$: Each such space is homeomorphic to some subspace of $I^\tau$. Compact Hausdorff spaces of weight not exceeding $\tau$ are homeomorphic to closed subspaces of the Tikhonov cube $I^\tau$. Thus, these two operations — taking topological products and transition to closed subspaces — suffice to obtain every compact space from a single, particularly simple, standard space — the interval. A remarkable consequence of the compactness of Tikhonov cubes is the compactness of the unit ball in a Banach space equipped with the weak topology. The universality of the Tikhonov cubes, and the simplicity of their definition, makes them important standard objects in general topology. However, the topological structure of the Tikhonov cubes is far from trivial. In particular, the cube $I^\mathfrak c$, where $\mathfrak c$ is the cardinality of the continuum, is separable, although it contains $2^\mathfrak c$ points; its weight is $\mathfrak c$. A surprising fact is that the Suslin number of each Tikhonov cube $I^\tau$ is countable, independently of $\tau$, i.e. every collection of pairwise-disjoint open sets in $I^\tau$ is countable. Although a Tikhonov cube contains many convergent sequences, these do not suffice to directly describe the closure operator in a Tikhonov cube.

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For references see Tikhonov theorem.