Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a continuous non-linear map, and let $A$ be a connected subset of $\mathbb{R}^n$ with $\text{dim}(A)=d\leq n$. When can we say that the dimension of the image, $\text{dim}(F(A))$, is also $d$? In other words, when does the map $F$ preserve dimension?


A non-example would be some sort of non-linear projection that embeds a set into a lower dimensional space.


I think that if $F$ is injective then this would be sufficient, but it also seems that we could weaken this significantly. For instance, maybe something along the lines of $F^{-1}(y)$ has at most countable cardinality for all $y\in\mathbb{R}^n$, although I am not sure how to guarantee this for a non-linear map.


If $F$ were measure-preserving then I think this would also be sufficient, but again somewhat overkill. In my case it is fine if $F$ changes the measure, so long as it doesn't change the dimension.



Dimension-preserving non-linear map
*Dimension Theory* by Ryszard Engelking provides the answer I was looking for. Specifically, theorem 1.12.8 which I have provided below, although there are other results in this same area that may be useful to someone.


**Alexandroff's Theorem**: *If $f:X\to Y$ is an open mapping of a locally compact seperable metric space $X$ onto a separable metric space $Y$ such that $|f^{-1}(y)|\leq \aleph\_0$ for every $y\in Y$, then $\text{ind}(X)=\text{ind}(Y)$.*


Here, $\text{ind}$ is the small inductive dimension, which for separable metrizable spaces is equal to the Lebesgue covering dimension (i.e. the topological dimension).


