The temperature limiter is a TMV or Thermostatic Mixing Valve. It will have two inlets and one outlet. One inlet is for hot, and the other is for cold. So, it would appear that what you have is two colds and one hot. The cold water pipe is probably split somewhere before you see it coming out to supply the cold side of the faucet and the TMV. The inlet and outlets are normally marked somehow, and it should be apparent if the outlet is going only to the removed fixture or continuing on to another fixture as well.
You should be able to cap off all the pipes without affecting any other fixture if the outlet only supplies to removed fixture.
In some cases (in the US anyway, it appears you are in London) the hot water can come up to one fixture and then go back into the floor and continue on to the next fixture. In this case, the two hot water pipes would need to remain connected to each other, but it doesn't seem like this is what you are seeing.
Based on your new picture: Yes, the output of that is only supplying that one sink. You should be able to remove the limiter and cap all the pipes coming up out of the floor. Mark the hot and cold pipes somehow to help out the next guy.