Tychonoff cube

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

The Hilbert cube, ${\displaystyle I^{\aleph _{0}}}$, is a special case of a Tychonoff cube.

The axiom of choice is assumed throughout.

• The Tychonoff cube is compact.
• Given a cardinal number ${\displaystyle \lambda \leq \kappa }$, the space ${\displaystyle I^{\lambda }}$ is embeddable in ${\displaystyle I^{\kappa }}$.
• The Tychonoff cube ${\displaystyle I^{\kappa }}$ is a universal space for every compact space of weight ${\displaystyle \kappa \geq \aleph _{0}}$.
• The Tychonoff cube ${\displaystyle I^{\kappa }}$ is a universal space for every Tychonoff space of weight ${\displaystyle \kappa \geq \aleph _{0}}$.
• The character of ${\displaystyle x\in I^{\kappa }}$ is ${\displaystyle \kappa }$.

• Tychonoff plank – the topological product of the two ordinal spaces ${\displaystyle [0,\omega _{1}]}$ and ${\displaystyle [0,\omega ]}$, where ${\displaystyle \omega }$ is the first infinite ordinal and ${\displaystyle \omega _{1}}$ 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.

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