On the page 28 of the book "An Introduction to Quantum Field Theory" by Michael E. Peskin and Daniel V. Schroeder in the last paragraph it says:
"When $(x-y)^2<0$ we can perform a Lorentz transformation taking $(x-y)\rightarrow -(x-y)$. Note that if $(x-y)^2>0$ there is no continuous Lorentz transformation that takes $(x-y)\rightarrow -(x-y)$"
Why is this happening?
Note that the metric signature is (+ - - -), as given in the preface/introduction of the book. Now let us take an example, which we can later extend to all frames related by Lorentz transformations.
Take $x$ and $y$ to be events lying on the time axis in some suitable frame. Note that
$$ (x-y)^2 = (x^{\mu} - y^{\mu})(x_{\mu} - y_{\mu}),$$
therefore in this frame $(x-y)^2 > 0.$ Now there is certainly no Lorentz transformation taking $t \to -t$, where $t$ is along time axis. We can now generalize this to all frames related by Lorentz symmetry, to generalize this for all $(x-y)^2 > 0,$ i.e. for all timelike events.
Similarly we can show the same for spacelike separated points too, lets say on either sides of the origin on the x-axis. The spacelike separated points have $(x-y)^2 < 0.$ We can show easily for a Lorentz frame moving with +v will have an opposite sign for $(x-y)$ in his frame than for a frame moving with -v, and both can be related by a continuous Lorentz transformation. Again, we can now generalize this to all frames related by Lorentz symmetry, to generalize this for all $(x-y)^2 < 0,$ i.e. for all spacelike events.
