Topology Vs. Real Analysis
Here's the dirt on how these objects relate to one another:
Some basic topology is absolutely essential to understand real analysis, to the extent that most textbooks on analysis actually include a whole chapter on topology. Concepts like "open interval", "limit point", and so on, of which we learn naive versions in calculus, need the machinery of abstract topology to generalize. Some topological notions like "simply connected" (naively "connected with no holes") look deceptively simple and yet without some rather advanced topology, it's almost impossible to rigorously define them.
Conversely, the real line and Euclidean space provide so many natural examples in topology that a topology textbook without lots of analytical examples would be like a barren wasteland.
Conversely, the real line and Euclidean space provide so many natural examples in topology that a topology textbook without lots of analytical examples would be like a barren wasteland.