A sufficient and necessary condition in terms of the compactness of super-distance space is given based on the property of super-distance space and the related theory of compactness.

By the property of the super-distance space and the connectedness of topological space,we obtained that all of the super-distance space its subspaces and product spaces are neither connected spaces nor arcwise connected space,meanwhile the super-distance space which isn t discrete topological space isn t partially connected space.

The property of a class of special distance space called super-distance space is studied,including its spherical neighbourhood,cauchy sequence,connectedness,completedness, etc.

In this paper,the Cartesian Product of three topological spaces,compact space,connected space and A2(A1) space,were studied,and three corresponding conclusions are given.

By the property of the super-distance space and the connectedness of topological space,we obtained that all of the super-distance space its subspaces and product spaces are neither connected spaces nor arcwise connected space,meanwhile the super-distance space which isn t discrete topological space isn t partially connected space.

In this paper, a characterization of paracompact, locally compact and connected spaces is given, and an example which shows that a connected and first-countable space can not be a continuous image of a paracompact, locally compact and connected space is constructed.

In this paper k-connected spaces are introduced and characterized.