continuous bijective maps between manifolds

Question

If $f$ is a continuous bijective map between two manifolds $M$ with dimension $m$ and $N$ with dimension $n$. Why this map does not necessarily preserve the dimension i.e $m\not = n$? What is a good example about that?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$Let $M$ be an arbitrary $n$-manifold with $n \geq 1$, and let $M'$ be the same underlying set with the discrete topology (an uncountable $0$-manifold). The identity map (on sets) $M' \to M$ is a continuous bijection from a $0$-manifold to an $n$-manifold.

Similarly, the $(n + k)$-manifold $\Reals^{n + k}$ may be written as the bijective image of the $n$-manifold $$ \bigcup_{x \in \Reals^{k}} \Reals^{n} \times \{x\}, $$ with $\Reals^{n}$ viewed as the prototypical $n$-manifold and $\Reals^{k}$ treated as a discrete manifold.

For a more interesting example, write a $2$-torus as a disjoint union of uncountably many real lines immersed as translates of an irrational winding.

Answer 2

In the case $n=m$ : Any injective continous maps $\mathbb R^n \to \mathbb R^n$ is an homeomorphism on its image. This is the classical invariance of domain theorem.

Since an immersion is not an embedding in general, my argument I did wrote is not true for $n < m$, as George Elencwajg said in the comments.