example of continuous bijections which are not homeomorphisms

Example 1. Assume that $X$ is a topological space, which neither discrete nor antidiscrete. We will show that there are topological spaces $Y$ and $Z$ such that there are continuous bijections $X\to Y$ and $Z\to X$ which are not homeomorphisms.

Let $Y=Z=X$ as a sets but topology on $Y$ is antidiscrete and on $Z$ is discrete. Then obviously identity mappings $\mathrm{id}:X\to Y$ and $\mathrm{id}:Z\to X$ are continuous, but since $X$ is neither discrete nor antidiscrete, these mappings are not homeomorphisms.

Example 2. Consider the function $f:[0,1)\to S^1$ (here $S^1$ denotes the unit circle in a complex plane) defined by the formula $f(t)=e^{2\pi it}$. It is easy to see that $f$ is a continuous bijection, but $f$ is not a homeomorphism (because $[0,1)$ is not compact).