Zero dimensionality and total disconnectedness

A totally disconnected set in a space is 0 dimensional (topological dimension) iff the space is compact.

can someone of you please give me an example where a totally disconnected set is not 0 dimensional?????

also after reading a few books on this i am a little confused as to what constitutes-

1) separation of point and another point .. is it closed disjoint sets containing them?

2) separation of two sets .. is it open sets being disjoint and union is whole space making them clopen.

3) separation of two sets by a set A .... is it finding a separation of X-A ? where X is the space?

Please help me..