<![CDATA[Metric space topology is a foundational yet challenging topic in undergraduate mathematics. This systematic literature review examines how students learn metric space topology by synthesizing research on cognitive processes, learning obstacles, and instructional approaches. Following PRISMA 2020 guidelines, we searched Scopus, Web of Science, and Google Scholar for publications from 2015-2024. Two reviewers independently screened titles, abstracts, and full texts. The final analysis included 38 peer-reviewed articles. Data were extracted using a standardized framework and analyzed through thematic analysis. Four major learning obstacles emerged: challenges in translating intuitive understanding to formal definitions; difficulties in treating mathematical operations as formal objects (within APOS theory); gaps between mathematical terminology and student reasoning (commognitive perspective); and challenges in understanding and constructing proofs. Analysis revealed five phases in typical cognitive development: (1) applying procedures from calculus, (2) becoming aware of underlying structures, (3) developing topological reasoning, (4) integrating formal definitions with examples, and (5) abstract thinking with proof-based reasoning. Findings suggest that instruction may benefit from: connecting formal mathematics to students' existing understanding, providing scaffolded proof instruction, and explicitly developing mathematical language and concepts. These insights may inform course design and pedagogical approaches in advanced mathematics]]>
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