In this paper, we investigate several conditions pertaining to closed range

multipliers on topological algebras. We first obtain some general results 

which give several equivalent conditions for a continuous linear operator 

$T$ on a Fr\'{e}chet locally convex space to have a closed range. 

In particular, when we assume $T$ to be a multiplier on a topological algebra 

without order, a number of other conditions also appear.

For instance, if $T$ is a multiplier on a semiprime Fr\'{e}chet locally 

convex algebra $A$ such that $T^2A = TA,$ then the range $TA$ is closed.

Finally, as a converse result, it is shown that if $A$ is a Fr\'{e}chet locally $C^*$-algebra and $T$ a multiplier on $A,$ then $TA$ is closed, if, and only if, $T^2A =TA.$