In this paper

the structures of five topologies on product spaces are discussed. A structural representation of the inductive limit topology is given by means of the Hamel base of a linear space

and another structural representation of the induced topology is given as Well. Then the relationships between the five topologies are discussed. Among these topologies the one- ordering topology is put forward by the author himself. In accordance with the mutual relationships between the five topologies on pro duct spaces

some properties concerning these topologies are discussed in this pader. The author also introduces the concept of maximal compatible subspace with respect to a topology on a linear space

which becomes a topological linear spaece

and the maximal compatible subsbaces are given to the box topology

the one-ordering topology and the induced topology on propuct spaces respectively

The completeness of the maximal compatible subspace with respect to the three topologies on the product space is given in a unified form. In this paper

the relationships between the normed topology

the projective limit topology

and the inductive limit topology on the Limit spaces

and also the relationships of the relative topologies of these topologies on the subspaces of the limit spaces are discusseo with theaid of structure. On the other hand

by taking advantage of the discussion on the completeness of the maximal compatible subspace with respect to the one-ordering topology

the proof of the sufficiency of the condition in the main proposition (4.1) cf [8] is much simplified.