Let $M$ and $N$ be $CW$-complexes.

**Definition. (different from the isotopy notion in geometry of submanifolds).** A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times [0,1]\longrightarrow N\times [0,1]
$$
such that $F$ maps the fibre $M\times t$ homeomorphically onto a subset of the fibre $N\times t$ for each $t\in [0,1]$. Two injective continuous maps
$$
f_1,f_2: M\longrightarrow N
$$
are said isotopy equivalent, if there exists an isotopy
$$
F: M\times [0,1]\longrightarrow N\times [0,1]
$$
such that
$$
f_1=F(\cdot,0),\\
f_2=F(\cdot,1).
$$
The $CW$-complexes $M$, $N$ are said isotopy equivalent, if there exist injective continuous maps
$$
f: M\longrightarrow N,\\
g: N\longrightarrow M
$$
such that their compositions
$fg$ is isotopy equivalent to $Id_N$ and $gf$ is isotopy equivalent to $Id_M$.

**Question.** Suppose $M$ is a compact manifold without boundary. Does there exist any non-compact $CW$-complex $N$ such that $M$ is isotopy equivalent to $N$? I cannot figure out any such example...