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 that 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 kind of sample have a technical name?"