Trnková was born on March 16, 1934 in Berehove, then in Czechoslovakia and now in Ukraine; her father was a forester. By the time she was in high school, her family lived in Prague, and she went to Charles University for study in mathematics. There, she worked with Miroslav Kat?tov on general topology, earning a master's degree in 1957 with the thesis Collectionwise normal and strongly paracompact spaces on strengthened definitions for normal spaces.
She continued her work on topology at Charles University as a doctoral student of Eduard ?ech, earning a candidate's degree (the Czech equivalent of a Ph.D.) in 1961 with the dissertation Non-F-Topologies. Much later, in 1989, she was also given the Dr.Sc. degree, corresponding to a habilitation.
In 1960, while still a student, Trnková became an assistant professor at Charles University. She was promoted to docent (associate professor), senior researcher, and full professor in 1967, 1986, and 1991, respectively.
Despite beginning her career working in general topology, Trnková shifted as early as 1962 to category theory. Her work in this area included studied of formal completions of categories, the embeddings of categories into categories of topological spaces, category-theoretic automata theory, and the isomorphism of product objects in categories.