By the Borsuk-Ulam theorem we know that every continuous map $M:S^n\to \mathbb{R}^n$ has a point in its range that is the image of two points in domain. by considering this I am curious to see what amount of this double point would exist for such maps and what we can say about the set of points in the range which are the image of one or odd number of points in the domain for all maps.
It is known that for every continuous Non one element range map $f_1:S^1\to \mathbb{R}$ there are properties which state that for an even $e\in \mathbb{N}$ the set of points in its range which their preimage have exactly $e$ points is uncountable and for evey odd $o\in \mathbb{N}$ the o- point set in codomain $\mathbb{R}$ has countable elements.
Also intuition says something similar but more complex about two dimension continuous maps $f_2:S^2\to \mathbb{R}^2$ which the dimension of $f_2(S^2)$ is $2$ (the same as codomain $\mathbb{R}^2$ dimension). It states that the o-point sets in $\mathbb{R}^2$ which $o\in \mathbb{N}$ is odd have one dimentional loop structure and for at least on even $e\in \mathbb{N}$ the e-point set has a two dimensional disc structure for each of its connected components.
Questions:
First: can anyone present more rigorous statements about the latter case above?(with proof)
Second :In general does there any invariant of the continuous maps relating to aggregation of characteristics info (structure, size, dimension ,...) of $n$-point ($\forall n\in \mathbb{N}$) and uncount-point sets of the map?
Note: $n$-point set for a map $f:X\to Y$ is the set of all points such $p\in Y$ that the preimage $f^{-1}(\{p\})$ consists of exactly $n$ points in $X$, if the preimage $f^{-1}(\{p\})$ consist of uncountable points in the domain then we call it a uncount-point.