For simple random sampling, I know that the probability of each point being part of the sample should be equal. Also, any sample of size say $k$ should be equally likely. In the sampling procedure I am using right now, the probability of each point being part of the sample comes out to be the same. However, if a point is chosen, some points are then more probable to be part of the sample. My question is, how important is the second criterion for the sample to be called simple random sample, and does such a sample have a technical name?