Tychonoff cube

Let $I$ denote the unit interval $[0,1]$. Given a cardinal number $\backslash kappa\; \backslash geq\; \backslash aleph\_0$, we define a Tychonoff cube of weight $\backslash kappa$ as the space $I^\backslash kappa$ with the product topology, i.e. the product $\backslash prod\_\{s\backslash in\; S\}\; I\_s$ where $\backslash kappa$ is the cardinality of $S$ and, for all $s\backslash in\; S$, $I\_s\; =\; I$.

The Hilbert cube, $I^\{\backslash aleph\_0\}$, is a special case of a Tychonoff cube.

The axiom of choice is assumed throughout.

• The Tychonoff cube is compact.
• Given a cardinal number $\backslash lambda\; \backslash leq\; \backslash kappa$, the space $I^\backslash lambda$ is embeddable in $I^\backslash kappa$.
• The Tychonoff cube $I^\backslash kappa$ is a universal space for every compact space of weight $\backslash kappa\; \backslash geq\; \backslash aleph\_0$.
• The Tychonoff cube $I^\backslash kappa$ is a universal space for every Tychonoff space of weight $\backslash kappa\; \backslash geq\; \backslash aleph\_0$.
• The character of $x\backslash in\; I^\backslash kappa$ is $\backslash kappa$.

• Tychonoff plank – the topological product of the two ordinal spaces $[0,\backslash omega\_1]$ and $[0,\backslash omega]$, where $\backslash omega$ is the first infinite ordinal and $\backslash omega\_1$ the first uncountable ordinal
• Long line (topology) – a generalization of the real line from a countable number of line segments [0, 1) laid end-to-end to an uncountable number of such segments.
• Maharam's theorem - a statement that complete measure spaces can be decomposed into unit intervals and discrete parts.

• Ryszard Engelking, General Topology, Heldermann Verlag, Sigma Series in Pure Mathematics, December 1989, {{ISBN|3885380064}}.

Category:General topology