Does an infinite number of zeros equal R1?

[student34]

I ask this because it seems that there is no distinction between 0 and an infinitesimal. Similarly, it also seems that an infinite number of one dimensional lines can equal R2, and the same seems to go for R2 to R3, and R3 to R4.

I only know basic calculus, so I am probably generalizing the concepts and not understanding the finer details.

 

[phinds]

An infinite number of zeros is zero.

Zero and an infinitesimal are not the same thing. If they were, we wouldn't need another name (infinitesimal).

I have no idea what you mean when you say

an infinite number of one dimensional lines can equal R2

 

[pwsnafu]

I ask this because it seems that there is no distinction between 0 and an infinitesimal.

If we are talking about the real numbers, there is no such thing as an infinitesimal. An infinitesimal is usually taken to be from outside of the real numbers (usually hyperreal).

Similarly, it also seems that an infinite number of one dimensional lines can equal R2, and the same seems to go for R2 to R3, and R3 to R4.

There is no "similarly". This is a completely different thing to infinitesimals.

A horizontal line on $\mathbb{R}^2$ is defined as
$L_a := \{ (x,y) \in \mathbb{R}^2 \,:\, y=a \}$.
Then it is obvious the (uncountable) union of all the $L_a$ where a is taken over all of the reals, is indeed $\mathbb{R}^2$.

Similarly, if x is a real number, the (uncountable) union of the singleton set $\{x\}$ taken over all of the real numbers, is exactly R. This is one of the most trivial things in set theory: if you have a set $S$ and take all its elements, then union them all, you get $S$ back.
But it has nothing to do with calculus or infinitesimals.