Abstract mathematical topics, like topology, can initially be challenging for students. Further, there is a limited amount of literature available for how undergraduate students learn topology concepts. This work aims to contribute to the literature by investigating the nature of students' difficulties in topology. I employed Skemp's (1979) model of intelligence and schema theory to construct a framework for successful action in learning topology, which has the potential for application in a variety of areas of advanced mathematics. Possession of an appropriate schema (Knowing That), a path from the present state to the goal state (Knowing How), and adequate skill (Being Able) are all required for successful action to occur. When the action is physical, it is directed by a delta-one director system. When the action is mental and is changing the state of a delta-one director system, it is part of a delta-two director system, which is where intelligent learning occurs. When the quality of the schema is improved, the functioning of the director system is also improved. A number of task-based interviews were conducted with a variety of participants: an undergraduate student, first-year graduate students, advanced graduate students, and a topologist. The framework was used to analyze the participants' activities when working with open sets and a basis for a topology. The results showed that undergraduate students can still be in the early stages of basis schema development, even at the end of their topology course, but the first-year graduate students showed significant learning outcomes by the end of their course. When the participants had issues with completing the problems, their difficulties occurred all throughout the Knowing That, Knowing How, and Being Able stages. Unlike the topologist, the novice participants showed a lack of strength of conceptual connections, less organization of their schemas, and had issues in all three stages for successful action.