Relation between Map and Dimension I am curious about two questions below
Let $M$, $N$ be two topological manifold.


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*If $\dim M>\dim N$, is there exist an injective continuous map $f: M\rightarrow N$?

*If $\dim M<\dim N$, is there exist an surjective continuous map $f: M\rightarrow N$?



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*To the first question, I think we can construct a map $F: M\rightarrow N\times\mathbb R^{m-n}$, which satisfies $F(p)=(p,0)$. If $f$ is injective, then $F$ is injective too. Then from invariance of domain, we can get $F$ is an embedding. Also, $f$ is an embedding. So is there exist embedding from $M$ to $N$?

*Second question can be transfered $N$ as $\mathbb R^n$ by using local coordination. Then I have know idea.

Any advice is helpful. Thank you.  
 A: For the first question, the answer is no, there is no continuous injective map from a larger manifold to a smaller one.
Suppose there existed such an $f$.  Then restricting $f$ to a chart $U\subseteq M$ we get an injective map $f|_U:U\rightarrow N$.  By further restricting $U$, we may assume $f(U)$ is a subset of a chart $V$ on $N$.
Thus, by composing with chart maps (which are homeomorphisms, and therefore injective), we may assume $f$ maps an open subset of $\mathbb{R}^m$ to a subset of $\mathbb{R}^n$ (with $m = \dim M$ and $N = \dim n$).  Abusing notation, I'll still write $U$ for this open subset of $\mathbb{R}^m$.
Now, consider the map $g:U\subseteq \mathbb{R}^m$ to $\mathbb{R}^m$ given by $g(x) = (f(x), \vec{0})$ with $\vec{0}\in\mathbb{R}^{m-n}$.  This will also be injective with image a subset of $\mathbb{R}^n\times \{\vec{0}\}$.  By Invariance of Domain, $g(U)$ is an open subset of $\mathbb{R}^m$.  But every open subset which contains a point of the form $(f(x), 0)$ also contains a point of the form $(f(x), v)$ for some tiny $v\in \mathbb{R}^{m-n}$.  This contradiction implies no such $f$ exists.
For the second question, yes, there are examples.  Perhaps the most famous are the Peano curves.  With some technical details sorted out, I believe one can prove that every (connected) topological manifold is the continuous image of $\mathbb{R}$.
