Uhh, no, they are closed and timelike. It's in the name.
The whole point of CTCs is that in exotic spacetime geometries, in particular wormhole-like structures, there can be curves that are entirely timelike, meaning that the object never moves faster than light locally, and are closed at the same time, meaning the object ends up at the same point in time from where it started.
In SR, a closed curve requires at least some spacelike sections of the curve on which the object moves faster than light.
The whole point of CTCs is that in exotic spacetime geometries, in particular wormhole-like structures, there can be curves that are entirely timelike, meaning that the object never moves faster than light locally, and are closed at the same time, meaning the object ends up at the same point in time from where it started.
In SR, a closed curve requires at least some spacelike sections of the curve on which the object moves faster than light.