Magnets have both north and south poles. You cannot remove one and keep only the other. Any piece cut off from a magnet becomes its own magnet having north and south poles. So in nature you will never find a magnetic monopole (north or south pole by itself) since magnetic poles always occur in pairs.
To know precisely why magnetic monopoles are impossible, you must understand the physics of magnetic fields. The magnetic field `vec(B)` is our physical interpretation of the circulation or “curl” in the vector potential field `vec(A)`. Mathematically it is expressed as `vec(B) = grad xx vec(A)` or “B equals curl of A”. This means all the field lines of the vector potential curl into closed loops, and the magnetic force field points perpendicular to this curl. Really there is no such thing as magnetic field in itself, it is just that a magnetic particle released in a curled vector potential field will be forced to travel perpendicular to the curled parts of that field, and this bundle of forces we label a magnetic field.
What defines a magnetic monopole is that its magnetic field diverges outward from a point source like the barbs of a sea urchin. Divergence of the magnetic field is a mathematical quantity represented by `grad cdot vec(B)`. Written in terms of `vec(A)` the divergence is `grad cdot (grad xx vec(A))` — which is always zero. It is mathematically and physically impossible for the curl of something to have a divergence. Therefore, magnetic fields cannot diverge from point sources, and hence there are no magnetic monopoles.
But what about the vector potential? Could there be such things as vector magnetic potential monopoles? Yes, definitely. Instead of magnetic field lines diverging from a source, it would involve vector potentials diverging from (or converging upon) a source. It would simply be represented by `grad cdot vec(A)` and since `vec(A)` is itself just the gradient field of a scalar “superpotential” field, mathematically it can very easily have a divergence. So while magnetic monopoles do not exist, magnetic vector potential monopoles could be very real. Strangely though, magnetic monopoles still get considerable research attention in the field of modern physics, while vector potential monopoles are not even publicly acknowledged.