Tychonoff cube $I^{m}$

Question

Different references have different definitions for the Tychonoff cube. For example;

• R. Engelking, General Topology: The Tychonoff cube of weight $m‎‎‎\geq‎‎‎‎\aleph‎_{o}$ is the space $I^{m}$, i.e., the Cartesian product $\prod\nolimits_{s \in S} {{I_s}}$, where $I_{s}=I$ for every $s\in S$ and $|S|= m$.

• K. Rao, Topology: Let $\mathbb{R}$ have the usual topology and $I$ be the closed unit interval $[0,1]$ have the relative topology as a subspace of $\mathbb{R}$. Let $X$ be a topological space and $F$ be the family of all continuous functions from $X$ to $I$. Let $I^{F}=I\times I\times I\times \ldots$ , where the number of factors equals the cardinality of $F$. $I^{F}$ is called a Tychonoff cube.

With both of these definitions can be shown that The Tychonoff cube is universal for all Tychonoff spaces, but Do these two definitions are equivalent?

Thanks.

Answer 1

Yes, they are. In fact they are exactly the same. A Cartesian product $\prod_{s \in S} X_s$ is just the set of functions $f$ from the index set $S$ to $\cup_{s \in S} X_s$ such that $f(s) \in X_s$ for all $s \in S$. So Rao and Engelking both describe the exact same set: all functions from some index set (whether it be $S$ or $F$; only the cardinality matters, topologically) to $I$, also denoted $I^S$, in the product topology, which is by definition the smallest topology such that all projections $\pi_t: \prod_{s \in S} X_s \rightarrow X_t$ (for $t \in S$), defined by $\pi_t(f) = f(t)$, are continuous.

The fact that Rao chooses $F$ specifically (it is itself a set of functions) is irrelevant. In fact any set $I^S$, whatever $S$, is called a Tychonoff cube. It's an easy exercise to see that when $S$ and $T$ have the same cardinality, $I^S$ is homeomorphic to $I^T$, so like I said before, only the size matters topologically.

Answer 2

As Henno Brandsma said, the Tychonoff cube depends only on the cardinality of the index set, namely the set $S$ in Engelking's definition and $F$ in Rao's comment. There is, however, this difference: Engelking allows $S$ to have any infinite cardinality you want. In the text you quoted from Rao, $F$ is the set of continuous functions from some space $X$ to the interval $I$, and the cardinality of this is not arbitrary. For example, it cannot be countably infinite.

I don't think Rao intended the text you quoted to be a definition of the terminology "Tychonoff cube"; that is, I don't think he intended to restrict to the special index sets $F$ and their special cardinals. I think he was merely commenting that the space he's using is one of the things called Tychonoff cubes, not that they are the most general such things. (This is why, in the first sentence of this answer, I wrote "Rao's comment", not "Rao's deifnition".)