confusion with an example in Munkres

Question

I'm currently studying algebraic topology and in the chapter on the fundamental group, he gives this example for a covering map.

I can't understand how $E$ can consist of $n$ disjoint copies if they are literally copies of the same set.

Answer 1

Note that $E\times\{1,2,\ldots,n\}=E\times\{1\}\cup E\times\{2\}\cup\cdots\cup E\times\{n\}$ and that$$i\neq j\implies E\times\{i\}\cap E\times\{j\}=\emptyset.$$

Answer 2

It means that $E=\bigcup_{i=1}^{n} X\times \{i\}$. That is, $E$ is the disjoint union of $n$ copies of $X$, since elements of $X\times \{i\}$ are of the form $(x,i)$ for $x\in X$, so $(x,i)\neq (y,j)$ for all $i\neq j\in \{1,\cdots,n\}$.

Answer 3

In math, we often call some things copies of other things if they are the same up to isomorphism. What an isomorphism is, depends on the category of things we are working with. In your case, we are working with sets, there isomorphisms are just bijective functions. As it was already pointed out, for $i\in\{1,\ldots,n\}$ holds $X\times\{i\} = \{(x,i):x\in X\}$. Can you see why there exists a bijective function between $X$ and $X\times\{i\}$?

The copies are made disjunct because for $i,j\in\{1,\ldots,n\}$ and $x\in X$ we have $(x,i)\neq(x,j)$ if and only if $i\neq j$.

Answer 4

They are not literally the same set. In math, the (somewhat informal) term "copy" refers to an object with "the same structure" as the original, but which is not literally equal to the original (much like a photocopy has the same content as the original page, but is on a new sheet of paper).

In this case, the $n$ "copies" are the sets $X\times\{1\},X\times\{2\},\dots,X\times\{n\}$. Each one is in bijection with $X$ (and even homeomorphic to $X$, using the product topology), but is not equal to $X$, and they are all disjoint from each other.