Definitions of Sierpinski Carpet and Higher Dimensional Analogues We define the Cantor Set as:
$Let \mathscr{J} := \{ 0, 2, \ldots , 3^{m-1} -1 \}$ for $m \in \mathbb{N}$, then 
$$C = [0,1] \setminus \bigcup_{m \in \mathbb{N}} \bigcup_{k \in \mathscr{J}} \Big( \frac{3k+1}{3^{m}} , \frac{3k +2}{3^m} \Big)$$
is it proper to define the sierpinski carpet and menger sponge, and their higher dimensional analogues as $C \times C$, $C \times C \times C$, and $\prod_{i \in \mathbb{N}} C_i$ respectively?  
I cannot seem to find an explicit formula for higher dimensionl analogues on the internet. Additionally, I'd like to know what it would mean to discuss the cantor set iterated a hyperreal number of times. i.e. 
$Let \mathscr{J} := \{ 0, 2, \ldots , 3^{m-1} -1 \}$ for $m \in \mathbb{N}^{*}$, then 
$$C = [0,1] \setminus \bigcup_{m \in \mathbb{N}^{*}} \bigcup_{k \in \mathscr{J}} \Big( \frac{3k+1}{3^{m}} , \frac{3k +2}{3^m} \Big)$$
Any feedback will be greatly appreciated, especially on the second portion of the question.
 A: It is easier to generalize what is removed in the cantor set.
The Sierpinski carpet is 
$$S = [0,1]\times [0,1] \setminus \bigcup_{m \in \mathbb{N}} \left \{ \bigcup_{k,l \in \mathscr{J}} \Big( \frac{3k+1}{3^{m}} , \frac{3k +2}{3^m}\Big)\times \Big( \frac{3l+1}{3^{m}} , \frac{3l +2}{3^m}\Big)\right \} $$
The Menger sponge is similar but more complicated, as we remove pieces on the edge.
Your second construction has a simple description. The regular Cantor set is the set of all ternary numbers between 0 and 1 that have a representation using only the digits 0 and 2. Your set is the same, but allowing the ternary representation to have a non standard integer number of digits.
For instance, for a nonstandard $m$, $$ \bigcup_{k \in \mathscr{J}} \Big( \frac{3k+1}{3^{m}} , \frac{3k +2}{3^m} \Big)$$ is the set of all ternary numbers all of whose representations have a 1 in the $m$th 'ternary place'.
But decimal or ternary numbers need care in hyperreal numbers:http://www.cut-the-knot.org/WhatIs/Infinity/9999.shtml
