We attempt here some characterization of near-rings and near-ring groups with some sort of finiteness conditions and topological vigor. In case of a topological ring, addition and multiplication are continuous on the product spaces, yet the coordinate wise continuity is all that is necessary in many cases. Here, in some results, we try to give some innovative look at the notion of what may be called so called "pseudo nilpotency" in case of an N-group. The same is dealt with keeping in note the corresponding Goldie character of any N-group. The so-called pseudo character on nilpotency and strongly semi-primeness, lead us satisfactorily towards our goal. in case of a topological group, where the binary operation is continuous in the product spaces, the corresponding coordinate wise continuity is an obvious characteristic in so-called one sheet space! But, the converse needs a hard work in the real sense if it happens in so-called broken two sheets space. Keeping aside the concrete so-called topological aspect of what has been explained above, we review this aspect of above type of algebraic structure from more or less algebraic point of view in a broaden courtyard .