One-to-one mapping and Topology preserving

Question

Does one-to-one mapping preserve topological properties ?!?, if then, how?

f: A (all sytstem states) -> B (one-to-one mapping; transformation), then A and B are topologically equivalent ?!?

Answer 1

The answer to your first question is negative. If $f$ is the identity map from $\mathbb R$ endowed with the discrete topology onto $\mathbb R$ endowed with the usual topology, then the domain is disconnected, whereas the range is connected.

Answer 2

Not necessarily. Take the interval $[0, 2\pi)$. If we look at the image in $\mathbb{C} \cong \mathbb{R}^2$ under $e^z$ then the image and the interval are topologically different. However there are cases where being one-to-one actually guarantees the same topological properties (I.e. guarantees being a homeomorphism). Take Brouwer’s invariance of domain. It says if $U \subset \mathbb{R}^n$ is open and $f: U \to \mathbb{R}^n$ is one-to-one (and continuous) then $f(U)$ is homeomorphic to $U$ i.e has the same topological properties