  This paper initiates the study of picture fuzzy topological spaces. In order
to develop a mechanism to construct picture fuzzy topological spaces, we prove
some basic results related to picture fuzzy sets together with the introduction
of many new notions such as balanced and \r{ho}-equivalent relations on picture
fuzzy sets. We also define the rank, picture fuzzy base and picture fuzzy
sub-base of picture fuzzy topological spaces. With the help of these notions,
we present a method to design picture fuzzy topological spaces, which is the
main outcome of this work. To the best of our knowledge, this is the first
serious attempt to study picture fuzzy topological spaces. It is hoped that,
this study will provide a decent platform for further development of picture
fuzzy topological spaces and their applications in decision making theory.
