This paper introduces several

weak continuity forms and graph conditions and provides some applications

of these forms and conditions.

The main tools which are utilized for their definitions are the

$\theta$-closure and u-closure operators. Applications

include (1) a characterization of quasi Urysohn-closed (QUC) topological

spaces analogous to that of the original definition of

Urysohn-closed spaces, i. e. that Urysohn-closed spaces are

those spaces which are closed subspaces of the Urysohn

spaces in which they are embedded, and (2) a weakening of the

continuity condition to improve the result that a

continuous image of a Urysohn-closed space is Urysohn-closed. Certain

subsets of a space are defined as quasi

Urysohn-closed (QUC) relative to the space and investigated.

Among the discoveries about these subsets is the fact that for a

Urysohn space there is a class of functions such that each

closed subset of the space is QUC if and only if each member of the class 

which

maps the space into a Urysohn space is a closed function. Parallels

for Urysohn-closed spaces of theorems for functionally compact

and C-compact spaces are provided. In the final section

of the paper the u-closure

operator is utilized to isolate a "second category type" property of

topological spaces.

It is proved that

QUC spaces have this property, and the property is employed to establish

several generalizations of the Uniform Boundedness Principle from

analysis.